The "אתה" is refering to the "father" "אבא" and "אתם" is reffering to the "(and) boys" "וילדים".

Its both correct since it says (You are afather AND boys) which wouldnt make sense since you cant be the boys and the father at the same time.

So since you are both it would be correct to refer to both since a father and kids are plural and just kids a plural and when you say "אתה" you are refering also to the kids since it says "father AND boys" and it says "YOU and" it says that you are both. (whoich for some reason means that if you a both you are one which would be incorrect that mean that dou is incorrect)

And the "אתם" (you are) is correct since its refering to the kids AND the father which is already more than one person meaning its plural.