Need ideas for a place to stay near Taormina with free parking

I would start the search by using something like booking.com and use the filters to find accommodations 'with parking' or 'free parking'

The beach town at the foot of Taormina's hill is called "Giardini Naxos". I assume there are lodgings there with parking. I didn't see anything as I arrived in town by train to make me think I wanted to spend some time wandering around there, but I imagine there's some older architecture somewhere. If you Google for photos Giardini Naxos, you'll get a lot of images. It's clear there's a beach there!

However, I have another suggestion, for a place I've never been. I know that sounds strange, but a Sicilian I chatted with while waiting for a bus recommended the town of Acireale. It's on the coast between Catania and Taormina, though much closer to Catania. Photos show a lot of historic architecture. It looks like my kind of place, but I didn't have time to get there on my trip in 2015. Again, Google for photos.

And, as already suggested, use a website like booking.com to find places in your price range with parking. Read all the reviews. I imagine at least a few people on this forum have stayed at Giardini Naxos, so you may well hear from someone with first-hand information yet. I'm not sure anyone has lodged in Acireale. But I've found the booking.com reviews quite accurate.

Edited to add:

Becky, I looked back at some of your other threads to see whether you mentioned Palermo, because I wanted to warn you about trying to drive into the city, and I came across your mention of wheelchair use. Are you aware that Taormina is built on the side of a steep hill? There are lots of steps between different levels, and during peak cruise-ship hours the streets can be quite crowded. You may be OK by going along the length of the main commercial street, but I'm not sure. I'm also not sure where you'd be able to get a taxi up to the Greek theatre, or how accessible the final approach is. It may be perfectly fine, of course. I suggest using Google Maps' satellite view and the little yellow man to walk some of the streets to get a feel for the terrain.

Some years ago we stayed at the Hotel Elios in Taormina, on via Bagnoli Croce. The manager (owner?) loaned us his pass to allow us to park free at a very close by car park building. Maybe ask him if he would be so kind?

Giardini Naxos is a good choice. There are hotels even in Taormina with parking included because it has lots of luxury hotels and people expect such an amenity. It's a good idea to look on booking.com. Are you planning on driving to Taormina (I assume you want to visit, right)?

I stayed at the hotel below years ago, with a car, and loved it. Loved the prices even more.
It’s in Giardini, on the beach, just 5 min drive from Taormina. They charge a small fee for parking, but not that expensive. I see that now the parking fee is 10€ a day, but rooms are inexpensive compared to Taormina up the hill, so it’s a wash.
If that is ok for you, the location can’t be beaten. You will have breakfast on a terrace facing the Ionian Sea in the morning. If you book a room facing the sea you will enjoy it even more.

http://i53479.wixsite.com/hotellariva/home-c18hn

Thank you all for the ideas! We’ll check them out. Becky

One other thought. We're staying at Residence Schuler this summer: http://www.residenceschuler.com/en/index.html. The prices for Claudio's apartments are awesome and they have wonderful views. The building is in Taormina proper. He offered us parking at 7 Euros a day - he has two small spots to rent out.

His apartment prices start about 105 per night and you can just add the 7 Euros to that. Not sure where that falls in your budget but we found it to me a great deal in expensive Taormina.