Shell, do you think they would be open to a conversation like . . .
"I know you must be really busy with the kids and all, but I've really fallen in love with your adorable kitty. Do you know if he's had his shots?"
If they say, "Yes, he's had his shots", then you've opened the door to more conversation about cat care, and you can mention that "he looks very thin. Has the vet said anything about that?" If they say, "No, what shots?" then say "Oh, cats need annual shots. I'm taking our cats in tomorrow and I can take Gizmo in too. It's really no trouble at all and I would pay for it etc etc"
y'know what I mean? Just sort of gauge what you say, by how receptive you think they might be to the idea of offloading the responsibility onto you, or another person.
Or something like,
"my friend Amy was over and just went head over heals for Gizmo. We were wondering if he was your cat or if he was a stray because he spends a lot of time at our house."
then see their reaction. If they seem relieved to have one less responsibility, then keep offering to take more and more responsibility for Gizmo.
Or depending on the age of the kids, have a heart to heart with the kids. Show them the proper ways to take care of a cat (without overtly criticizing the parents) and encourage the kids to be Gizmo's advocate. When you talk with the kids maybe you can get a picture if the family would rather not have the pet as a responsibility anymore. Maybe they would be open to "sharing" Gizmo, etc.
Just a few ideas to think about. The bond between a human and an animal can be so strong, as you are experiencing now. I see so many people when a kitty is missing and want to prevent the same loss if I can. An intentional re-homing â€” even if for what you believe is a good cause â€” can be devastating to both humans and animals. You don't really know for sure that Gizmo is neglected do you? Maybe you're absolutely right in that the parents have been irresponsible, but if the kids are bonded to Gizmo, where would they be if they came home and they believed their kitty went missing or was dead?