"Rigid" or "fixed" batten parrels.
While on the subject of "rigid parrels" (for example, your "pistol parrels", Steve's "egg former" or "oval parrels" - and, for that matter, the arrangements used on Paul Mc's various aerojunks and hybrids - and (I think) David T's Weaverbird wingsail - these are all rigid versions of the conventional soft mast parrel, - we might as well also throw into the mix a concept which Arne was toying with a while ago.
I hope Arne doesn't mind.
Arne sent me a drawing of his concept, which he calls "fixed batten parrels", which he was thinking of trying out, but which later he found he did not need, so the idea "never left the drawing board".
In Arne's usual clever way, he worked out a solution, in two different forms:
Arne's "fixed" batten parrel (asymmetric)

Like the "pistol" parrels, this is an asymmetric form of Steve's "oval" parrels and Paul Mc's "D former" parrels.
There is quite a lot of "food for thought" in this forum thread.
Jan's "Pistol parrel" (asymmetric)

Steve's "egg former" or "oval parrel" (symmetric) for SJR

Paul Mc's "D-former parrel" (symetric) for SJR

Soft batten parrels
(a) Conventional soft batten parrels (long type or short type) do not need a diagram.
(b) Slieve's soft "spanned, paired, running downhaul parrels" for SJR

Soft batten parrels (a) and (b) are for the asymmetric system (the sail stays on one side of the mast on both tacks) - and it would seem that they work best on a sail planform which has been harmonised with regard to mast balance and yard angle, so that the halyard angle is within reasonable bounds.
The "rigid" or "hard" (fixed) types above would seem to be much more tolerant of planform and halyard angle, because the yard and battens are restrained from any fore-and-aft movement.