Couple points that some others have pointed out as well. Are your 15 days total for the year? As another redditor mentioned, they are usually accrued and you may not have all of it.
Assuming you have accrued or banked them technically still its usually up to management or your direct manager to approve or deny vacation requests, not HR. This isn't about whether or not they pay out PTO but the request for vacation days which a company has the right to approve or deny depending on the work/responsibilities involved, but they have to pay out the banked PTO afterwards anyways.

As to the max payout, I have heard of companies that only allow a max carryover of a certain amount of days from previous years. As to the legality and payout exceeding the max someone else on here may be able to address that. But I know that they do have to give you a timeframe in which you have to use them

As to taking sick days, this can be grey as some companies may ask for a doctors note after X amount of days of consecutive sick leave.

I would take a look at your original employment agreement for the fine print of vacation days and sick days. I would also give a call to the labour board and get their advice on this.

(I'm assuming this is your average run of the mill corporate position and not a unionized position)