Everyone is legally entitled to holiday leave (not pay - the two are different).
Thus, if you are legally employed (eg, not a cash in hand, no payslips, no tax arrangement), you are legally entitled to holiday .
This applies whether you are full or part time (although if you are part time then the amount of time off you get is pro-rata'd).
People in full time employment are entitled to 20 paid days off a year (which can be made up of bank holidays). If you worked half of the typical working week (eg, where full time hours consitituted as 40hrs and you did 20hrs) you would get 10 days holiday a year.
Your employer is NOT allowed to pay off the holiday days to you unless you are a) leaving the company and the holiday leave is due to you or b) work on an as and when basis so holiday is paid as you go (a lot temp agencies work this way). This is in pursuit of a healthy work life balance.
Your situation is difficult because if you are entitled to the time off, NOT the money. What I would do is go to your employer and say "I should have had 20 days holiday for every year - I have therefore worked out that you owe me X days off. I can either take these or we can reach a compromise where you pay me for some of them and I take the rest as time off".
In future, I would make sure you take the leave you are entitled to because your employer is not obliged to pay what you haven't taken, to you. effectively, if you don't use it you lose it.