The Avenger was Reggie Park. I know I have a photo of the Avenger from  
1974-75 somewhere.I will see if I can dig it out.
 
In Jim Wilson's book Mr. Wilson says he worked under the hood for Jim  
Crockett Sr. as The Avenger in a program against Ole Anderson with the
angle being he was avenging a past injury he suffered at the hands of Ole.
If this program happened as Mr. Wilson says,then I must have slept through
it. At the very least it never happened in Winston-Salem or Greensboro where
I attended most of the house shows I ever saw.
 
This would have had to take place before Jim Crockett Sr.'s death on April
1,1973 and Jim Wilson joining Ann Gunkel's All South Wrestling Alliance in  
Georgia.I can tell you this,I was at a card in Greensboro on Nov.2,1972 where
Jim Dillon and "The Avenger" were scheduled to wrestle Dick Shannon and
Evil Eye Gordon (Guillotine Gordon and later Enforcer Luciano in 1980). The
Avenger no showed and Terry Kay took his place.
 
I do think it is possible that Jim Wilson was one of several different wrestlers
who wore a mask and wrestled with the Anderson's as the "Menace".

This was one of my favorites as a kid.  The Avenger was the only other wrestler (besides Rufus R. "freight train" Jones) who could wiggle out of the Super Destroyer's claw.  On one of his earliest TV appearances, he got in the ring with Super D and every time he escaped the claw, he would reverse slam him.  I remember Super D leaving the ring and then commenting to the announcers about how bad his back hurt.  It was a great angle while it lasted.

Did Parks still call himself the Avenger after that, and wrestle without the mask, or was he just Reggie Parks after being unmasked?

I was at that Greensboro card and it was Parks under the hood.
But to be really honest, I always thought it was Doug Gilbert for years.
Gilbert must have done something masked/ unmasking because I see this listed many times.