Agree with north-north-west. Walking from Parachilna down to Quorn would take in some of the best parts of the trail. If you go as far as Melrose, this would also allow you to walk down up and down Mt Remarkable after spending the night at Gray's Hut (which is a great little spot).
My favourite part of this section (and of the whole trail) was actually walking from Mayo Hut to Hawker via the Yourambulla Range and Jarvis Hill Lookout. Walking along that range is as good as anything on the Larapinta in my opinion. Some people miss this by just taking the alternate routes into/out of Hawker (as most need to resupply) - the range is definitely worth the extra effort.
Aside from that, a few other tips for other parts, some slightly further south of the Flinders:
* Quorn - Maybe do the railway and the video show onto the silo was great (although make sure you rug up - it was freezing!).
* Walking along the dry stone wall up New Campbell Hill and then down to spend the night in Curnow's Hut was wonderful - the hut is in a lovely position
* I loved the section from Hallett to Caroona Creek - this includes Mt Bryan (especially the view from "God's Seat"), Mt Bryan East and Tourilie Gorge was a wonderful section to walk (the day after going south through those hills north of Burra not so much though)
* Burra - the town has heaps to look at, I think I walked further on my Burra "rest day" than any other day on the trail

Another thing I enjoyed were the old country pubs that were full of character. Ones like the Spalding Pub, the "Wild Dog" in Hallett and the Georgetown Hotel (the "pub that fell from the sky") all helped make my Heysen walk even more memorable.