I had the same problems when starting out but with a BXA size, piston style QCTP. From what I have gathered, the best solutions for parting involve replacing the QCTP with something more rigid mounted on the cross slide or compound, either in front or back, or inverted or not. The lack of rigidity of a QCTP is less of a problem for other types of cuts (turning, facing, boring) because there are ways to compensate that are not possible when parting.
Note that the no. 7 parting blade holder is sloped upwards and creates positive rake on the cut. This is suitable for a few materials but is prone to digging and jumping on most steels. There is no way a tapered blade can be ground to reduce the top rake so the no. 7 holder should be the first to go.
Parting blocks as mentioned in previous posts are level with the cut (neutral rake) and would be more suitable in some cases. However this does nothing to improve the QCTP itself.
Carbide parting tools tolerate chatter less than HSS and really only have their place once your act is together with HSS and you want higher production rates. Carbide is by no means a solution to chatter in a poor set up.
If your QCTP replaced a 4 way tool post (like those often supplied with a new lathe) it could be reactivated and dedicated to parting. A four way tool post is more rigid than a QCTP. One or more parting blades would only have to be mounted once in it. Then all you do is swap the tool posts on the compound when its time to part.
Ultimately the parting holder that mounts directly to the cross slide would have the most potential to be rigid. How far one is willing to go is a personal decision but parting with a QCTP and No.7 holder combination is not hard to beat.