No one can tell you over the internet what ammunition may or may not be safe in a given gun.
The Remington era Parker specification sheets shown on pages 164 to 169 of
The Parker Story clearly state 2 5/8 inch chambers intended for 2 3/4 inch shells. In theory, a 1935 vintage gun was made well after SAAMI specs for ammunition came into effect and your Parker should handle any commercial 2 3/4 inch lead loads. Whether the 86 year old wood is up to it is a whole different issue. Any of these guns that saw use in the 1920s to 80s probably digested plenty of Western Super-X, Peters High Velocity, Remington Heavy Duck Loads (later Nitro Express) or Winchester Super-Speed shells. These were the popular shells when your gun was made --
Late 1930s NITRO EXPRESS.jpg
High Velocity 12-gauge 2 3-4 inch roll crimp.png
REPEATER SUPER-SPEED 12-gauge 2 3-4 inch #4.jpg