The string portion (is that Picard's theme?) is indeed actually better than I remember.
No, Picard's theme is the part just before the strings. After the Courage fanfare, we get 8 bars of the main theme on horns, then 8 bars of Picard's theme on horns, then 8 bars of the main theme on strings, then a closing 4-bar statement of the main theme. The Picard-theme portion begins at just about one minute into the cue.
I won't go so far as to say it overall sounds like a parody but I will say that it reminds me of The Last Starfighter and other mid-80s genre orchestrations.
It sounds like Dennis McCarthy doing a John Williams riff, or rather a classic march of the type that Williams brought back into fashion with
Star Wars, but which has its roots in the film scores of the '30s and '40s.
As for the "parody" comparison, I think it's worth noting that the music in Mel Brooks movies was generally approached rather earnestly and was quite well-done, so I don't think that cuts it as a negative comparison. Maybe you think of that musical sound as comical due to the context, but taken on its own, it's not that different from serious dramatic scores from equivalent genres, albeit maybe a bit broader and livelier.
Ultimately, to me it feels like it tries really hard whereas Goldsmith's comes across as more effortlessly melodic and natural with the melody and rhythm having a more, as the pretentious would say, organic interplay whereas the brass portions of McCarthy's composition are halting and don't flow in a way I find appealing.
I can see what you mean about "trying really hard"; I have always found it a bit bombastic. But as for the melody, note that the second through fifth notes of the main motif are actually the same as the first four notes of the Courage fanfare. I think that's a nice touch, providing a connection to what came before.
Then again, I've heard Goldsmith's so many times (and, truth told, at times even been utterly sick of it) that not only would I need to listen to the McCarthy tune a couple dozen times to shake my preconceived notions completely out of mind but I'd have to forget Goldsmith's and learn it over again fresh.
Actually the main reason I find McCarthy's alternate theme interesting is because it's not the familiar one. As TrekBBS posters might remember from my past discussions about Superman themes, I like it when composers get to create new themes rather than just copying an old one. I'm not saying I don't understand why the producers passed on this particular theme; it is a little too big and brash and not an ideal fit for the show that TNG was. But I don't think it's a bad piece of music per se.