I have a lot of 12/2. I have a 20 amp appliance that I am replacing with a new 20 amp appliance. However the old appliance is 110 and the new appliance is 220. Can I run a second 12/2 romex to convert to 220? My understanding is that this is not paralleling (not allowed) where you try to handle a 30 amp load by using 2 20 amp (12 gauge) wires.
Paralleling is not allowed for branch circuits. You answered your own question there!
Now the good news, hopefully IF (the big IF) there is nothing else connected to your existing 120v appliance circuit, AND the new 240 v appliance doesn't require a neutral, you could repurpose the existing circuit by getting a double pole breaker and moving the white wire to one of the poles on the breaker, mark it with black tape making it a 240v circuit. There is nothing wrong or bad about running 240 on 12/g, it's commonly done. I believe most wiring for residential use is rated for 600v.
If the appliance is socket/plug connected it's imperative that the outlet/socket be replaced so no-one could plug a 120 volt device into the outlet. NEMA 6-20R would be the proper outlet/socket. If hardwired, no worries. Some might say 4 wire is required (2 hots, neutral and ground), but if the appliance doesn't require a neutral, you'll be fine. More details about the new appliance would help us give you a better answer.
It is not going to be acceptable to try creating a 220/240VAC circuit with two separate 12/2 Romex cables. The largest part of the load in a 220/240VAC circuit is current flow between the two HOT legs. You want to have these two legs in the same cable so that they work together to cancel out as much of the magnetic and electrical fields around the wires as possible. For this reason for a 12A 220/240VAC circuit you must use either a 12/3 with GND type Romex cable or resort to individual wire conductors routed through conduit or an approved raceway.