The super call must supply all parameters for base class. The constructor is not inherited. Commented out artist because I guess it is not needed when doing like this.

class StreetArtist extends Artist {
  constructor(
    name: string,
    age: number,
    style: string,
    location: string,
    public medium: string,
    public famous: boolean,
    public arrested: boolean,
    /*public art: Artist*/
  ){
    super(name, age, style, location);
    console.log(`instantiated ${this.name}. Are they famous? ${famous}. Are they locked up? ${arrested}`);
  }
}

Or if you intended the art parameter to populate base properties, but in that case I guess there isn't really a need for using public on art parameter as the properties would be inherited and it would only store duplicate data.

class StreetArtist extends Artist {
  constructor(
    public medium: string,
    public famous: boolean,
    public arrested: boolean,
    /*public */art: Artist
  ){
    super(art.name, art.age, art.style, art.location);
    console.log(`instantiated ${this.name}. Are they famous? ${famous}. Are they locked up? ${arrested}`);
  }
}

Another solution I eventually came up with, in addition to the ones provided by @iberbeu and @Nypan, is to add and intermediary initProps() method right before the call to scream():

class Person 
{
    public firstName: string;

    constructor(firstName: string, props?: any)
    {
        this.firstName = firstName;
        this.initProps(props);
        this.scream();
    }

    protected initProps(props: any): void
    {
    }

    protected scream(): void
    {
        console.log(this.firstName);
    }
}

class Employee extends Person
{
    public lastName: string;

    constructor(firstName: string, lastName: string)
    {
        super(firstName, {lastName});
    }

    protected initProps(props: any): void
    {
        this.lastName = props.lastName;
    }

    protected scream(): void
    {
        console.log(this.firstName + ' ' + this.lastName);
    }
}

Although I think both made a strong point and I should actually be using a factory pattern instead..