Why am I getting denied for credit though I paid off my debts?

Hello. I have gotten denied by 3 credit cards that are meant for students with limited credit history. The reason given is that I have delinquent accounts on my credit report. I pointed out that these are all PAID, but the companies still refuse to give me a credit card even when I said I was willing to start with a very low credit limit! These debts were not even my fault to begin with: they were medical debts incurred when I was 13 that my parents were too irresponsible to notice and address. I desperately need to build credit because I will be graduating next year and want to get a car and an apartment. But those accounts probably won't drop off my credit report for a few more years. How can I build credit?