Several years before any other PD symptoms I noticed that I kept sliding down if I sat up in bed and also on the sofa and as you describe sometimes minutes after I sat myself up. My mother-in-law had PD and this was something she had done but I dismissed this along with many other little signs as hypochondria. Since being on sinemet this does not happen much.
Certainly Pilates which targets core strength would be helpful if you could find a sympathetic teacher. For example, local to me there is a seated Pilates class