The way dataclasses combines attributes prevents you from being able to use attributes with defaults in a base class and then use attributes without a default (positional attributes) in a subclass.

That's because the attributes are combined by starting from the bottom of the MRO, and building up an ordered list of the attributes in first-seen order; overrides are kept in their original location. So Parent starts out with ['name', 'age', 'ugly'], where ugly has a default, and then Child adds ['school'] to the end of that list (with ugly already in the list). This means you end up with ['name', 'age', 'ugly', 'school'] and because school doesn't have a default, this results in an invalid argument listing for __init__.

This is documented in PEP-557 Dataclasses, under inheritance:

When the Data Class is being created by the @dataclass decorator, it looks through all of the class's base classes in reverse MRO (that is, starting at object) and, for each Data Class that it finds, adds the fields from that base class to an ordered mapping of fields. After all of the base class fields are added, it adds its own fields to the ordered mapping. All of the generated methods will use this combined, calculated ordered mapping of fields. Because the fields are in insertion order, derived classes override base classes.

and under Specification:

TypeError will be raised if a field without a default value follows a field with a default value. This is true either when this occurs in a single class, or as a result of class inheritance.

You do have a few options here to avoid this issue.

The first option is to use separate base classes to force fields with defaults into a later position in the MRO order. At all cost, avoid setting fields directly on classes that are to be used as base classes, such as Parent.

The following class hierarchy works:

# base classes with fields; fields without defaults separate from fields with.
@dataclass
class _ParentBase:
    name: str
    age: int
    
@dataclass
class _ParentDefaultsBase:
    ugly: bool = False

@dataclass
class _ChildBase(_ParentBase):
    school: str

@dataclass
class _ChildDefaultsBase(_ParentDefaultsBase):
    ugly: bool = True

# public classes, deriving from base-with, base-without field classes
# subclasses of public classes should put the public base class up front.

@dataclass
class Parent(_ParentDefaultsBase, _ParentBase):
    def print_name(self):
        print(self.name)

    def print_age(self):
        print(self.age)

    def print_id(self):
        print(f"The Name is {self.name} and {self.name} is {self.age} year old")

@dataclass
class Child(_ChildDefaultsBase, Parent, _ChildBase):
    pass

By pulling out fields into separate base classes with fields without defaults and fields with defaults, and a carefully selected inheritance order, you can produce an MRO that puts all fields without defaults before those with defaults. The reversed MRO (ignoring object) for Child is:

_ParentBase
_ChildBase
_ParentDefaultsBase
Parent
_ChildDefaultsBase

Note that while Parent doesn't set any new fields, it does inherit the fields from _ParentDefaultsBase and should not end up 'last' in the field listing order; the above order puts _ChildDefaultsBase last so its fields 'win'. The dataclass rules are also satisfied; the classes with fields without defaults (_ParentBase and _ChildBase) precede the classes with fields with defaults (_ParentDefaultsBase and _ChildDefaultsBase).

The result is Parent and Child classes with a sane field older, while Child is still a subclass of Parent:

>>> from inspect import signature
>>> signature(Parent)
<Signature (name: str, age: int, ugly: bool = False) -> None>
>>> signature(Child)
<Signature (name: str, age: int, school: str, ugly: bool = True) -> None>
>>> issubclass(Child, Parent)
True

and so you can create instances of both classes:

>>> jack = Parent('jack snr', 32, ugly=True)
>>> jack_son = Child('jack jnr', 12, school='havard', ugly=True)
>>> jack
Parent(name='jack snr', age=32, ugly=True)
>>> jack_son
Child(name='jack jnr', age=12, school='havard', ugly=True)

Another option is to only use fields with defaults; you can still make in an error to not supply a school value, by raising one in __post_init__:

_no_default = object()

@dataclass
class Child(Parent):
    school: str = _no_default
    ugly: bool = True

    def __post_init__(self):
        if self.school is _no_default:
            raise TypeError("__init__ missing 1 required argument: 'school'")

but this does alter the field order; school ends up after ugly:

<Signature (name: str, age: int, ugly: bool = True, school: str = <object object at 0x1101d1210>) -> None>

and a type hint checker will complain about _no_default not being a string.

You can also use the attrs project, which was the project that inspired dataclasses. It uses a different inheritance merging strategy; it pulls overridden fields in a subclass to the end of the fields list, so ['name', 'age', 'ugly'] in the Parent class becomes ['name', 'age', 'school', 'ugly'] in the Child class; by overriding the field with a default, attrs allows the override without needing to do a MRO dance.

attrs supports defining fields without type hints, but lets stick to the supported type hinting mode by setting auto_attribs=True:

import attr

@attr.s(auto_attribs=True)
class Parent:
    name: str
    age: int
    ugly: bool = False

    def print_name(self):
        print(self.name)

    def print_age(self):
        print(self.age)

    def print_id(self):
        print(f"The Name is {self.name} and {self.name} is {self.age} year old")

@attr.s(auto_attribs=True)
class Child(Parent):
    school: str
    ugly: bool = True

Define another method to initialize u and d, so that you can override that part of B without overriding how qu and qd are defined.

@dataclass
class B(A):

    u: float = field(init=False)
    d: float = field(init=False)
    qu: float = field(init=False)
    qd: float = field(init=False)
    
    def __post_init__(self):
        super().__post_init__()
        self._define_u_and_d()
        self.qu = (math.exp((self.r - self.div) * self.dt) - self.d)/(self.u - self.d)
        self.qd = 1 - self.qu

    def _define_u_and_d(self):
        self.u = 1 + self.pu
        self.d = 1 - self.pd



@dataclass
class C(B):
    def _define_u_and_d(self):
        self.u = math.exp(self.sigma * math.sqrt(self.dt))
        self.d = 1/self.u

Here is a working solution for python > 3.10

@dataclass(kw_only=True)
class TableMetadata:
    """
    - entity: business entity represented by the table
    - origin: path / query / url from which data withdrawn
    - id: field to be used as ID (unique)
    - historicity: full, delta
    - upload: should the table be uploaded
    """

    entity: str
    origin: str
    view: str
    id: str = None
    historicity: str = "full"
    upload: bool = True
    columns: list = field(default_factory=list)


@dataclass(kw_only=True)
class RestTableMetadata(TableMetadata):
    """
    - method: HTTP method to be used
    - payloadpath: portion of the response payload to use to build the dataframe
    """

    method: str
    payloadpath: str = None

I would argue that #1 is the most correct method. For the example you showed, it appears to be irrelevant which method you use, but if you add a second variable, the differences become apparent. This is implicitly confirmed by the Inheritance section in the documentation.

@dataclass
class ParentClass:
    a: str
    b: str = "parent-b"

# This works smoothly
@dataclass
class ChildClass1(ParentClass):
    a: str = "child-a"

# This works, but is a maintenance nightmare
@dataclass
class ChildClass2(ParentClass):
    def __init__(self, a="child-a", b="parent-b"):
        super().__init__(a, b)

# This works, but it changes the signature and only works if a is first
@dataclass
class ChildClass3(ParentClass):
    def __init__(self, a="child-a", **kwargs):
        super().__init__(a, **kwargs)

Right now, the dataclass decorator is adding default methods, including __init__ to your class. That means that if you wanted to use option #2 or #3, you would have to know and copy the function signature for all the parameters. At the same time, option #1 allows you to change the default for just a.

The other way to do what you're doing is to create a __post_init__ method for your child classes, which can then override the parent default value:

@dataclass
class ParentClass:
    a: str = ''  # Or pick some other universally acceptable marker

@dataclass
class ChildClass(ParentClass):
    def __post_init__(self):
        if self.a == '':
            self.a = "child-a"

This is also needlessly complex for most scenarios, but may be useful for a more complex situation. Normally __post_init__ is meant to be used to initialize derived fields, as in the example in the linked documentation.