They won't hurt the painted ladies unless the ladies are eating something they'll eat, and the ladies are much smaller than the tiger cats - they could accidentally eat them along with the leaf.
The favorite host plant for tiger swallowtails here is wild black cherry/Prunus serotina. Do you have a wooded area near you where there are tulip trees growing wild or wild black cherry? They'll also eat Ptelea trifoliata, but that's not too common, and they reportedly will eat sassafras.
The best thing to do would be to find some wild tulip trees and put them on them.
Sherry

Thanks you for the response, Sherry. It's good to know that the painted ladies will be okay. I wasn't sure how they would interact. Unfortunately, I don't think we have any of those plants, not even wild tulip trees. My parents had to buy ours, which is why they don't want the tiger caterpillars munching on it and perhaps damaging it. I've been trying to give the tigers willow leaves and branches, and while they seem to like climbing on them, I haven't noticed them eating any. They're sticking to the tulip tree leaf. Isn't there anything else? If there isn't, my mom will probably let us put them back, but I'm hoping for an alternative.

There is a long list of tiger host plants in one of my books.
They are:
Arizon ash
Carolina ash
Green ash
White ash
Black cherry
Common chokecherry
Common peach
Mexican plum
Apple
Camphor-tree
Common catalpa
American hornbeam
Eastern cottonwood
Mock orange
Sassafras
Sweet bay magnolia
Tulip tree
Common lilac
Hop tree
Spicebush

I hope you have some of these and one of them will do.

Sherry

Thank you so much! We have many, many wild lilac bushes around our house, so hopefully they'll do fine there, no worse for the wear. I really appreciate your help. =)