My Cat is Sick and I'm Scared

please help, I need advice. content warning for animal hurt/pain. I'm super stressed and angry right now. there's this stray cat in the park I feed and it has severe fur loss in the back to the point it's balding. it's scared of carriers and I can't ever catch it in time. my dad doesn't usually get home till after 5 and that's when the city shelter closes. there's another one we could go to but they are only open 3 days a week and close at noon. my dad won't take a day off to help me get it and when he gets home early enough he comes up with exuses as to why we shouldn't bring it. I feel so guilty and like I'm gonna be responsible if something bad happens. I have no other ride that can take me and idk what to do. I don't know how to contact animal control and if I'd have to pay or not for them to grab it. I'm also worried they'll put it in a kill shelter. on top of that my mom is being negative and coming up with reasons with why I shouldnt bring it to the shelter or call animal control and I don't understand why they are being such assholes. idk what to do im only 15.the only other person who can take me is my sister and she only has Sundays and I think Mondays off and Sundays the shelters are closed. please help I don't know what to do. I can't bring it in the house because I have two other cats and one likes to antagonizing.

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