The first cars that came to my mind were the 67 Impala's GM siblings, the Olds 88, the Pontiac Catalina, and maybe the Buick LeSabre, altho it's a bit higher end and hence not so much a competitor.I was wondering what cars other companies made to try and compete with the Impala.
The first Ford I thought of was the Galaxy 500, as Pauly noted. The Mercurys on that platform are, like the Buick, a bit higher end.
I'm kinda ignorant of the Chrysler products of that era, but the Plymouths were usually positioned in the market with the Chevy and Ford brands. The Fury comes to mind since it was a full size car, like the Impala.
Chrysler, like GM and Ford, built several other models on this same platform, including other Plymouth models such as the Belvedere, as well as Dodge and Chrysler branded versions. That is, when all the badge engineered variants are included, there were many competitors.
(As I wrote this, it was interesting to realize how the brands (in the American market) have changed over the years. Plymouth, Olds and Pontiac are gone now. If we were comparing contemporary cars, other brands such as Toyota, Nissan, BMW, et al, would be in the discussion instead.)
Anyway, when I was a kid growing up in the late 60's, the Plymouth, Chevy and Ford brands were on the same rung of the competition ladder, while Buick, Mercury and Chrysler were up a rung. Cadillac and Lincoln were higher still. I don't think Chrysler corp. back then competed well at that level, but the Imperial might be on that rung.
Looking at how these cars were marketed back then, I'd say the most direct competitors of the 67 Impala would have been the Fury and the Galaxy 500.
BTW, ever since I got my license in 1977, I've wanted to have a 66 Galaxy 500