This obviously depends on your employers, but as long as you've accrued sick time or time off, you and your partner should be able to supplement your parental leave with those hours. You most likely will not be able to stack the time off, though. For instance, if your employer pays you 50 percent of your salary while you're on parental leave, you can't take time off or sick time on those days to bring yourself up to full pay.
However, if you are entitled to four weeks of paid parental leave and you want to be off for five, you can use accrued paid time off for that last week, as long as your employer has approved it.