Op you can be stingy if you want but the normal expectation is that on snow or PTO days she is paid the same as if she were working.

In any other industry, your PTO is paid out in actual hours accrued which is usually dependent upon hours worked. In this case OT does not apply.
If you've set yourself up with an agreement that gives "one week" or "5 days" off (or however many days it is) rather than hours then yes, you should pay the regular weekly rate.

OP, your scenario is confusing, can you clarify how many hours the nanny worked in her week and what your PTO agreement is?

Weekly take home stays the same, whether vacation, sick time, or snow days.

What about snow days? Would you pay OT for snow days?

Legally, you only need to pay it as straight time. Practically? Reducing someone's paycheck will lead to a lot of resentment! If you want a good relationship, I'd consider the full weekly guarantee, straight + OT hours, as the minimum payment you make weekly.

Legally, you only have to pay OT on hours worked. That being said, if you have a guaranteed hours agreement with your nanny, she will expect to receive her usual pay even on weeks she uses PTO. I would be upset if my employers basically docked my paycheck whenever I used PTO. It would kind of defeat the point of PTO and OT and I wouldn't work for them for long.

Looking for advice on how you typically pay your nanny for paid time off.

Here is the factual circumstances I am considering:

Nanny typically works 50 hours per week.
Nanny rate is $20 for first 40 hours and $30 for OT hours.

Let's say the nanny takes 10 hours of paid vacation or sick leave one week.

Do you pay the 10 hours of vacation or sick leave at the regular rate or the overtime rate?

The nanny technically only worked 40 hours but I also can see an argument that if the time off is truly "paid" you add it to the hours worked and pay time and a half.

Advice? Thoughts?