TL;DR
The song is not in Dorian or Aeolian. It's built around the D minor pentatonic scale.
But to understand how the determination is made...
From the Department of Taking Things too Far
There is general agreement that the primary pitches are D, E, F, G, A, C. So, here's a detailed look at the other pitches that occur in the song.
B: 00:21 - 00:24. Guitar lick.
B-something: 00:50 - 01:01. (implied) In neither the G5 power chord, nor in the vocal harmonies it accompanies, do I hear a clearly articulated B or Bb; however, I hear the chord more easily as G than Gm; possibly because B occurs before Bb in the G overtone series.
B: 00:29 - 00:30. Same as 00:21.
Ab: 01:38 - 01:40. To be really picky about figuring out the mode, one must acknowledge the A flat in the guitar.
B: 01:43 - 01:45. Same as 00:21.
Ab and Db: 01:48 - 01:51. A variation on 00:21.
B and Bb (and Gb): 01:53 - 01:56. Another variation on 00:21 .
B: Passing notes in the guitar at 02:17 - 02:18 (ascending) and (maybe, but definitely not B-flat) 02:27 - 02:28.
B: 02:31 - 02:33. Guitar ascending passing tone.
This is essentially the same as the album version of part II, except for the clear modulation to F from 01:24 - 01:43, which oscillates between F and Bb chords (temporary I and IV) before moving to a transitional C (V/F and bVII/D).
Bb chords occur during the guitar solo in the same context as the album version, part II, but more frequently. For example, 06:05 - 06:19.
Finally, during the guitar solo, at 04:43, there's a passing Eb.
Conclusion
It's easy to see why the song would be interpreted as Dorian: the primary note selection corresponds to that mode. B predominates by far, so can be considered far more influential in determining the mode than Bb. All other pitches (Db, Eb, Gb, Ab) are rare and clearly do not contribute to determining the mode. (Note that the Db in part I, 01:48, is playing the role of Db and not acting as the leading tone C#.)
However ...
This is not Dorian. To be truly modal, one would expect to hear far more of the B to emphasize the modality. Further, it would occur in different contexts. B only occurs when accompanied by G. In a truly Dorian setting, we would hear, say, B against the F chords, or in more rhythmically or melodically accented positions. Further, the only times we hear Bb, it's clearly intended as a chromatic alteration (part I, 01:53) or part of a modulation to F.
It's also not Tonal Minor. To be minor in the Tonal sense, there would have to be the presence of a leading tone; i.e., C#. This never occurs (see above regarding the Db).
It's also not Modal Aeolian (natural minor). To be Aeolian, Bb would be essential, and B to be avoided.
Therefore ...
The song is decidedly built around the D minor pentatonic scale, with lots of E for color, and a fair number of G-B pairs just to make things confusing also for color, but occurring through modal mixture via D major, rather than as an indication a true Dorian.