Which was not introduced in any GM cars until the 1969 model year...

As Salmon stated, this transmission is clearly non-original to the car, which could likely result in confusion for you if you were searching for past problems with shifter buttons on 1967 Firebirds. There is nothing wrong with having a non-original trans, though (unless having a #'s matching car is a priority for you...which I hope it was not...cuz it seems yours is not.) I have a TH350 in my bird, too. Most would consider it an upgrade over the original 2 speed Powerglide your car was likely built with...depending on which engine it had. 400's got TH400s instead of the Powerglides that most of the smaller-engined 67's got.

I agree with your mechanic. Indeed there is no cable related to the shifter, just a linkage between the shifter and the trans. And for a TH350, if your car was properly converted when installed, there should also be a kick-down cable leading from trans to the carb...but realize this particular cable, while necessary for proper downshifts, has absolutely nothing to do with your shifter. So if the button is not holding it in park, indeed this should be a shifter issue, and not a transmission, or linkage, or cable issue.

I believe you'll need to do some shifter disassembly and inspection to really trouble-shoot it.