Which parent goes in “breed” & which in “crossbreed” when different?

Should I enter a foal’s “breed” according to his sire or his dam & then put the other’s breed under “crossbreed”?

My one foal has coloration most like his sire but if is ever entered in a parent/foal class he would go with his dam.

It does not matter which you put it under when it comes to entering classes - the class requirements take all breeds listed into account when you go to enter. If it is, for example, an Arabian x Quarter Horse cross, whether you put Arabian as the main breed and Quarter Horse as the crossbreed or Quarter Horse as the main and Arabian as the cross, you will be able to enter it in both the Arabian class or the Quarter Horse class because it has both of those breeds listed.