This question is much deeper than I originally understood. The question is:
Why do we need to refer to a parent scale at all when talking about the modes? All of the 7 modes are diatonic scales, so why not simply refer to them as the "diatonic modes" and leave it at that? And if, for some reason, we do need to refer to a parent scale, why do we choose the major scale to be the parent scale instead of, say, the dorian scale?"
We have to refer to a parent scale because, among other things, modes are a mechanism for creating new scales. We start by choosing an interesting and less common scale, like harmonic major, and we see what happens when we start on a different root. We realize, "hey, if we start on the 5th scale degree, it sounds great over a dominant 7th (♭9♮13) chord!" When we see our friend later that day, we say "try playing the 5th mode of the harmonic major scale in measure 3." Referring to the parent scale helps us communicate clearly. It's how people tend to think about those less common modes--in terms of more common parent scales.
Given how important the first mode is, it receives a special designation: we call it the "parent scale" for all of the other modes. But for the case of the diatonic modes, aren't all of them already well known? Yes, but there's still a reason why we don't choose dorian to be the parent scale/the 1st mode. The major scale has a very special status (even more special than other parent scales like harmonic major, melodic minor, etc.).
The major scale is special because we use it to define every other scale. Browsing through the names listed in this post, we see things like phrygian ♮6, lydian ♯2, and dorian ♭5. What do those numbers refer to? These numbers/alterations refer to the notes of the major scale. When we say:
natural minor scale has a flat third, a flat sixth, and a flat seventh
what we really mean is:
natural minor scale has a flat third, a flat sixth, and a flat seventh compared to the major scale
The major scale serves as our benchmark or baseline. That's why, given the 7 diatonic modes, we choose it to be the parent scale.