Hi JDK, I have walked the path you are going down. I had a gelding (morgan of course) quite a few years ago that had been the typical A-rated Morgan saddleseat (SS) and driving horse. Now he wasn't super hot, but he could be...

I showed him saddleseat, hunter at morgan A shows and schooled him to 2nd level dressage...the saddleseat self-carriage was actually quite helpful, since he already knew how to work off of his hind end. We progressed to schooling 2nd level quite fast, in about 6months.

I'm sure you know this stuff, but here's what worked for us: I had an excellent instructor that appreciated SS for its pros (vs cons) and she was taught by Sally Swift (personally!) and some famous French dressage guy (I'm no DQ, so apologies lol) named Francois. We did alot of ground work to teach him correct lateral aids and how to stretch his neck w/rein pressure (this was key), and learning to respond to minimal rein pressure. Once he learned that rein pressure now meant stretch, vs pop up into the bridle, it was smooth sailing... The other things that made him "think" vs going into his jiggy walk, were lots of shoulder in and corner/shoulder work in general...this horse, and most SS horses are never taught to really turn w/roundness - so getting him to use his shoulders and simultaneously stretch into that bit forced him to work long/low and "think" through it, vs charge through the corners. The other best thing for a hyped up ex-SS horse is ground driving - he already knows how to do it, most likely, and it allows him to learn w/out getting all hyped up about a rider on his back.

BTW - the parking out will eventually disappear once gets on the bit and LISTENS and waits on your next cue...it may take a few months, but before you know it, it will be gone. Most of my morgies only do it when they are excited and charged up.

I do have a mare now that I've used the same training methods, and she is a more hot morgan...both parents were national/world champ park/halter horses...so I knew what I was getting into. She has responded well to the same methods, but tends to get over the bit/rushy/jiggy as well when she gets excited...what worked best for her was Clinton Anderson method of "throwing away the reins" and then doing a one-rein stop w/minimal pressure. This calms her nerves and allows me to get her back on the bit w/out a blowup. Very counterintuitive since when you feel the horse tensing up/getting hot, we tend to brace ourselves and tighten the reins...but throwing them away is the best thing I've learned to keep in my toolkit.

Your gelding is very cute, barouqe and round - I don't see any real confirmational faults.
The shoulder angle is traditional morgan...made for up/down movement, but he will def be able to extend/swing out once he understands how to use his body. It won't be warmblood-esque, but extended w/more loft. These horses can really move once they learn what's correct.
Sorry I rambled on and on...it sounds like you're having a great time w/him! Keep us updated!