The older Parkers fitted with DiMarzios are fairly bright instruments- and you will not find an instrument with better playability. The newer ones fitted with Duncans are a bit beefier by comparison.
Overdriven, I find the Parker's bridge a bit thin, but using the neck pickup fattens up the tone substantially. Honestly, it's no brighter than a Tele and I'm sure it's well suited to the tones the LS generates. Parkers do clean tones like nobody's business and through an amp like the LS, it should really be quite beautiful- even with mild overdrive, I reckon.