If you are in the states you should accumulate sick days by law even if you work part time, I believe it’s one hour for each 35 hours so you should have a full day after 2 months or so. Is it a lot? No. But it has to be provided to you. Well he can’t demand you’ll go inpatient when you are off work. Ideally we’d do major medical things when we are off but things happen when they happen