Hotel Iberia, round the corner from plaza America, in a narrow street with a building site next door
Very central, and a basement garage for the bike/car
Hotel Don Carlos. A 5 minute walk to the main plaza. A small comfortable and reasonably priced hotel. The parking is about 200 years away, they have some spaces in a private secure parking lot. Drop your bags off first. Booking.com
I’ve used the Gran Hotel Don Manuel twice cancelling the Parador the second time to stay there since it was cheaper. Minutes away from center by foot. Underground parking and vast breakfast.
Hotel can be accessed front and back. Walk out of rear for center, parking entrance on other side via the very open and modern main entrance. Do not confuse the two.
We used the Parador in the Old Town the last time we visited, in an old 14th c Palace. IIRC, the secure parking was off-site.
It's a fascinating City with a magnificent historic centre, but it gets blistering hot in Central Spain in the summer (>40c), the stone/granite buildings are still hot to the touch at midnight.
I'm not clear about the OP's dates/ itinerary, but the Salamanca or Plascencia Paradors probably make for a more relaxing experience, if that matters, with a pool & secure parking on-site.
Thank you to everyone who replied. I've booked a night at the Hotel Iberia Plaza America, cheaper than Don Manuel which was my second choice.
Both have secure underground car parks.