I'm missing a few subtleties, but it's basically:
Except for two measures, the solo is entirely chromatic phrases, and I think this is significant. There's a piece of classical music called Einzug der Gladiatoren, Op. 68 by Czech composer Julius Fučík. It's often used in conjunction with circuses, and the well-known section is also mostly chromatic. Here's an excerpt:
The solo in "Palisades Park" uses the same musical vocabulary, and I think the intent is to suggest the carnival atmosphere with which Einzug der Gladiatoren is often associated. This is similar to the type of environment that the lyrics of "Palisades Park" describe.
I also referenced the version of "Palisades Park" by Freddy Cannon, which I think is the original and which the liner notes of my edition of 15 Big Ones mention specifically. This feature is there too, but in the Beach Boys' version, it's used to a greater extent and is much more prominent.