Parking is still free at TI, and there's still the tram that runs from essentially the TI parking garage to Mirage, which helps a little with the amount of walking to get around. The garage is very large, and I wouldn't anticipate it being full except for very busy weekends on the strip.
As far as I know, the parking garage at Planet Hollywood is still free, as it's the parking for the Miracle Mile Shops. While it's a hike through the shops to get to the casino, I'd venture to say that's your closest free option for Bellagio.