String myString = "1234";
int foo = Integer.parseInt(myString);

If you look at the Java documentation you'll notice the "catch" is that this function can throw a NumberFormatException, which you can handle:

int foo;
try {
   foo = Integer.parseInt(myString);
}
catch (NumberFormatException e) {
   foo = 0;
}

(This treatment defaults a malformed number to 0, but you can do something else if you like.)

Alternatively, you can use an Ints method from the Guava library, which in combination with Java 8's Optional, makes for a powerful and concise way to convert a string into an int:

import com.google.common.primitives.Ints;

int foo = Optional.ofNullable(myString)
 .map(Ints::tryParse)
 .orElse(0)

parseInt is the way to go. The Integer documentation says

Use parseInt(String) to convert a string to a int primitive, or use valueOf(String) to convert a string to an Integer object.

parseInt is likely to cover a lot of edge cases that you haven't considered with your own code. It has been well tested in production by a lot of users.