To identify the peroxide, we analyze the oxidation state of oxygen and the nature of the oxygen anion in each compound:

1. BaO₂ (Barium Peroxide): Barium is an alkaline earth metal (Group 2) with a fixed oxidation state of +2. For the compound to be neutral, the $O_2$ group must have a charge of -2 ($O_2^{2-}$). This is the definition of a peroxide ion. ($Ba^{2+} [O-O]^{2-}$).
2. KO₂ (Potassium Superoxide): Potassium is an alkali metal (Group 1) with a +1 charge. The $O_2$ group has a -1 charge ($O_2^-$), which characterizes a superoxide.
3. MnO₂ (Manganese Dioxide) & NO₂ (Nitrogen Dioxide): These are normal dioxides where oxygen exists in the -2 oxidation state ($O^{2-}$). For example, in $MnO_2$, Mn is +4 and each O is -2.

Therefore, BaO₂ is the only peroxide among the choices.