Not so fast. Or, as they say, don't believe everything you read on the internet.
Thanks for all this theory Doug. So after all my first guess about the "A drone" was not so wrong
I disagree with keeper's post on so many levels that I was going to just sit on my hands and not say anything, but I feel I need to keep others from getting drawn in.
The words "tonic" and "dominant" do not refer to notes, but to chords. During the medieval period, when the plagal modes had relevance to the way music was put together, nobody thought in terms of chords.
The fact that the note A held as a drone throughout the tune sounds better than D does not make it a tonic. It just means that most of the tune is harmonized with A chords and D minor chords, both of which contain A, but only one of which contains D.
I still maintain that the tune is in D minor (look how it ends), not dorian or any other mode. The fact that it starts on a dominant chord is unusual but unheard of. To convince yourself it's not dorian, think about what chords you would harmonize it with. The first half of the 4th measure seems to cry out for a G minor (with a Bb) rather than a G major (with a B natural). The B natural in the melody is just a byproduct of the way melodic minor scales work, and is not part of the mode.
I could go on and on, but I think we'll just have to agree to disagree.