I just put one of the VX Freedom 2-7x33 Rimfires on a Marlin 60. Sighting it in at 25yds, parallax was an issue above about 5X. I then pushed the target back to 100 and maxed out magnification to 7X and had zero issues with parallax.
I'm pretty big into rimfires and also an optics geek. In the past I've had one of the Leupold 6.5-20 EFR Rimfire Target scopes. Trying to squirrel hunt with that thing was a nightmare. Everyone talks about how great the EFR (Extended Focus Range) scopes are for rimfires, but outside of a bench with known distance I've found them to suck because they're just too sensitive. Like, the difference between setting the parallax adjustment on 10yds and 20yds is enough to give you trouble with the shot.
Generally speaking, I just use the "Rimfire" Leupolds on .22's. They're supposed to be parallax free at 60yds and I've not had issues with them out to 200yds. On my 10/22 Target model, I replaced the Leupold 6.50-20 EFR with a Leupold 4.5-14 Tactical AO. It has the capped target style turrets and the AO goes down to 25yds, which works well enough for me. Unlike the EFR scopes, which have multiple revolutions to the AO, the standard AO is only about 180 deg revolution to go from 25yds to Inifity and as such, setting it somewhere in the middle (100yds) is perfectly useable for most field conditions, but it can be dialed down if shooting at close range.
In your situation, I'd sell both those EFR scopes (they bring a lot right now), but the 2-7x33's on the centerfires and pick up a VX-Freedom 2-7x33 for the rimfire.