Yes, the Bazooka brand was always part of Topps. Now, precisely what the parent company was named, I do not know off hand. However, I know for certain that Topps and Bazooka are from one and the same parent company.
As you look at many of the photos used on the run of Bazooka baseball issues, Topps either initiated the photo with a Bazooka card, or re-used a photo from one of their mainstream regular Topps set, or even a test issue.
For example, the 1965 Bazooka Mickey Mantle photo was later seen in the 1969 Topps Super Baseball test issue. Both great cards.
Back to football, you can really tell the similarity in the setting for the Johnny Unitas photo shoot, with the nicest portrait going to the 1959 Bazooka, and the obvious Johnny mugging a "half-crazed" pose selected for the 1960 Topps release. Wish I owned a '59 Bazooka Unitas.
All the best. ---Brian Powell