The current Adcom amps are good, not great. At least the 555MS, 555SE and 565SE. I haven't heard the 575SE or the 585SE. The design is tried and true, and they have better parts as you move up the line. I would think that the Parasound Classic line is pretty close in performance.
I currently use the Adcom GFA 565SE, and with my speakers I can't get it to heat up whatsoever. 88 db sensitive speakers, Kef Q750. The 565SE shares the same topology of the much bigger 575SE, and I expect one would have to push it pretty hard to approach thermal protection.
The sound has good detail with a slight nod towards warmth. I'd like to try it out with some higher end speakers, maybe one day. No bells or whistles on these Adcom amps, you don't get the trigger or signal sensing turn on until you pony up for the 575SE. I did have an Emotive XPA-3 amp previously and the Adcom 555 and 565SE definitely are an improvement over that.
The Parasound Halo line is on another level, which is to be expected at the prices.