When you mean 'cut your mom off' do you mean completely or just from the pups? Can't you go alone to visit them (without the pups)?
I personally think that your father is right. If hygiene issues can cause physical stress on a 70 year old, then I wouldn't risk taking the pups over there even if it's only a few times a year. Do you feel like the pups may cause your mom's eye problem to worsen?
As far as respecting both people's wishes, sometimes you can't. Taking sides with parents is hurtful to the parent that is on the opposite side and a pride boost to the one whose side you're on, but when there's a right and a wrong, you do have to make a choice...
But you can be diplomatic about it. Maybe saying something as simple as 'I love you and I'd feel guilty if my pups caused your eye problems to flare up' might do. So instead of making it seem like you're taking a side, you present the solution in such a way that you're feelings are the center of attention...
I dunno...that's JMO.