Neighbor dropped 7-8 cats in our backyard, what do I do?

Hello! I am writing this post for my parents. Our neighbor (we live in a duplex) recently got told that she had to move out because her landlord sold her side of the home to a family. Turns out, her daughter had taken in a litter of kittens & raised them, but since they moved to an apartment, they abandoned them and left. They are about 6-8 months old now maybe. Ever since, these cats have been roaming around the neighborhood, and I don't know what to do. They are very friendly, not aggressive at all, and seem to like the indoors, but we prevent them from entering the house since we own a single cat of our own and don't want to spread fleas. My parents have been doing their best to feed them very moderately as to encourage them to roam around and find someplace else to go, but they refuse to leave our backyard and they sleep all day since they're weak and there's too many to feed. I feel so lost. They also are not spayed or neutered at all, so they will only multiply. Do we call a shelter and trap every single one ourselves? I also hear that many shelters are overcrowded and they usually euthanize. Are there any organizations in Houston that can help? We live around Sugar Land, but our address is still classified as a part of Harris County. Does anyone have any advice for this?