2 pole 15 amp breaker off of a 2 pole 20 amp breaker
Yes, you can change this branch circuit into a feeder.  Feeders supply subpanels.  

Note that feeders do not supply branch circuit loads, which means your existing 20A load will need to be added to this subpanel, weird though that may be. 

So your new subpanel will need:  2 spaces for a 2-pole breaker for existing 20A/240V load, and 1 space x 2 for the 120V/15A circuits.  They don't need to be on a 2-pole breaker, unless they share a neutral. 

Now, they do make 2-space/4-circuit panels, but those rely on "double-stuff" breakers which are obsolete because they do not support AFCI, which has been required on most circuits since NEC 2014.  Also, some panels do not make the special breaker you'd need (double tied quadplex).  Because of all this, you need to plan a full space for each circuit. Beyond that, you may also want expansion room. Here's what you need in a panel. 


Bus rating >= 20A. (This won't be a problem)
Main breaker is required if this is in an outbuilding. Must be >= 20A
Main breaker optional otherwise.  Must be >= 20A 
Spaces >= 4 (often marketed as >= 8 circuits)
Accessory ground bar (not included in every panel) *


A 6-space/60A is fine.  A 40-space/200A is fine.  Whatever you have lying around. 

You must separate neutral and ground in the subpanel. 

You attach the existing 12/3 cable to the ground bar, neutral bar, and 2 main lugs or main breaker. 

Fit a 20A 2-pole breaker in this panel, and run cable to your existing load.

Fit either a 15A 2-pole breaker, or two 15A single breakers.  Attach your 120V loads there. 

Neutral and ground go on separate bars.  Make the neutral wire long enough to reach the breaker, as this will be required for any future AFCI/GFCI installation.  



*. Actually in your case, because of the very small size of the panel, you can simply wire-nut all neutrals together and have no neutral bar at all.   That lets you re-task the bar in the panel to be the ground bar.  No accessory ground bar needed! This would not work with plug-on-neutral panels.