As usual its most helpful to actually analyze specific examples.
The only Allman songs with harmony lines I had handy were a couple of old ones: Revival ("people can you hear me... love is in the air") and Dreams.
Here are the intervals between the two guitar lines from selected sections, divided by bar lines. "4" = interval of a fourth, "5" = interval of a fifth, etc.
Revival intro
4 | 5 | 4 | 3
Revival: Dm section (about 0:55)
note: I've notated based on what meter sounds like to me here: two bars of 3/4, followed by a bar of 4/4, repeat
3 | 3 | 6 5 | 5 | 5 | 4 |
Dreams: outro
5 | 6 | 3 | 4
Maybe what Dickey meant when he said this:
>
was that a lot of times if you approach playing a harmony part you tend to think in intervals of thirds or sixths, rather then fourths and fifths, which from the (admittedly small) sampling above, seems to be what the Allmans sometimes favored.