You have some difficult choices ahead, with no easy path forward.
Trying to manage this home situation while being a resident is going to be very difficult. You'll need to be at work for 10-12 hour days, with minimal phone interruptions. You may have some flex time to go to appointments, but only at a reasonable frequency. If those appointments are more than monthly, you may find this challenging. You're going to be working multiple night shifts in a row and need to sleep during the day -- will that be possible?
One of the most important attributes of a resident (and a physician for that matter) is reliability. As a medical student you're completely disposable. As a resident you're a critical piece of infrastructure. You can't just leave work for an hour, or spend lots of time on the phone. If there are frequent emergencies it's going to be a huge problem. Your parent is going to need to care for themselves for long periods of time, and you'll need to be comfortable with that.
As others have mentioned, if no one else is willing to care for them, then you can always get them in a nursing home. It will not be easy. You need to separate your finances -- figure out what assets they have, and which are yours. I find it hard to believe that you don't qualify for Medicaid -- they are disabled and likely have minimal income, and you're a medical student with no income. Regardless, it doesn't matter -- they could be placed in a nursing home, they spend down their assets -- all their money goes to paying their stay. Until they have nothing left, and then they qualify for Medicaid. And that pays for the NH forever.
But this is really hard, and goes against your culture. You basically need to "divorce" your parent. There's no legal requirement to financially support your parent. They may hate their life in a nursing home, and you'd need to live with that. Then again, perhaps they might be much happier there with other people around.
Waiting until you match is a prescription for disaster. Getting someone into a nursing home, if that's what you plan to do, takes months. Medicaid applications take months. Medicaid is a state based program -- so if you have medicaid in one state it's a whole process to try to get it in another. The match is in mid March, and your new job starts in June. There's absolutely no way anyone will be able to help you settle this in that amount of time.
And matching to be near relatives? From what you describe here, that sounds like a pipe dream. They are not interested in helping, moving closer will not help at all.
There may be some fields, and programs, that may be more flexible. But doing a field you don't enjoy is certain to make you miserable too. And no field is easy -- all involve long hours that you may have little to no control over.
I do wish you the best. I highly recommend you discuss this with mentors at your school. If your culture has a religious component to it, getting help there may be an option also. Ultimately, even if he is put in a nursing home while you're a resident, you would then be able to afford to get him out and back in your home if you want with supports. If you are unable to complete a residency, then you'll both be stuck.