That's interesting. I wonder if the Strat HR is different than the Tele HR. They're hot, but not TOO hot. And I find them bright. Not dark. Que sera sera.
I still have the bridge Tele HR around. My Esquire is certainly not dark acoustically either. It's REALLY thin and wiry acoustically, even. It's got a pine body and a maple neck and fretboard, and a Gotoh bridge. I did even wire the HR through a single 500K Volume knob, not tone knob, so it's certainly not the setup, I think.
But I remember I'd plug my LP's in with a Black Winter and an X2N each, and then the Esquire, and that would sound so dull by comparison. Like no attack and aggression at all. It didn't sound muddy because the Tele HR is not particularly bassy either. I really wanted to like that pickup, but I just couldn't jive with it, even when playing metal (which is honestly all I ever play, LOL). That pickup is all output and no tone, IMO.
I never tried to run it in parallel, though. Maybe I would've liked it, but honestly, I doubt I would, considering I didn't like it split either.
My perception is that it wouldn't pair well with the Jazz Neck, personally, because the HR is like the complete opposite EQ profile, and then if the OP wants balanced output between pickups, the HR is just going to DESTROY the Jazz output-wise, IME, and then he'd be left with the same problem, just the other way around.