"The Party" is on the 1967 Phil Ochs album, Pleasures of the Harbor. There's a YouTube recording of the song here (click).

I hear an additional word after "she steals a glance at lovers" in the "Wallflower" verse, but I can't make out what it is.

All I get out of the shrugging shoulders/piano tuning is that Ochs is an unnoticed and uninvolved observer. To me, the piece is reminiscent of Spoon River Anthology.

-Joe-