Hotel/Parking in Edinburgh

We were hoping to stay close in and walk to everything, but the close
in stays don't have parking.

Yes, that's why the general advice is to not have a car if you plan to visit Edinburgh, or any other European city for that matter.

If we stay further out, we still have to find parking in old town
area,

Why would you need to find parking in Old town if you're staying further out? That doesn't make sense.

will that be a problem?

It would be, parking in Old town is very rare.

If you stay further out, look for a hotel with easy access to the tram or bus. Then you won't have to worry about driving into Old Town Edinburgh and finding parking.

Is it an option to drop off your car when you arrive in Edinburgh, or to pick it up when you depart?

Here is one way to skin this cat - The tram from the airport to the center of town stops at Edinburgh Park . fifteen minute easy ride is right at a short walk from a Premier Inn and a Novotel , both good hotels., with adequate reasonably priced (Novotel ) and free parking ( Premier Inn ). look here at the map https://www.google.com/maps/place/Premier+Inn+Edinburgh+Park+(Airport)+hotel/@55.9272505,-3.3122536,16.75z/data=!4m19!1m8!3m7!1s0x4887b800a5982623:0x64f2147b7ce71727!2sEdinburgh,+UK!3b1!8m2!3d55.953252!4d-3.188267!16zL20vMDJtNzc!3m9!1s0x4887c431c92b0363:0x6c6f477e42c1ea24!5m3!1s2024-03-02!4m1!1i2!8m2!3d55.928075!4d-3.308192!16s%2Fg%2F1hc2gmyg2?entry=ttu

We often stay at the Holiday Inn Express in Leith, which has parking. Good bus link from there into the city centre. Bonus of being very close to the Royal Yacht Britannia if that's something you'd want to visit while in Edinburgh.