Since a Dread would be a creature within 10 feet of itself, would it be affected by its own aura?
The short answer is that NO, it does not cancel out their own immunity. As people have pointed out, this is a problem beacause "aura" has not been defined in the rules.
We will add errata to our powers and abilities to denote this, so that things become clear.
Aura of Fear states:
At 3rd level, a dread radiates a palpably daunting aura that causes all enemies within 10 feet to take a –4 penalty on saving throws against fear effects.
As the character is not her own enemy, then no, this does not affect her.
An Aside about RAI:
I suppose you could read the following sentence in isolation of context and come up with the idea that it does:
Creatures that are normally immune to fear lose that immunity while within 10 feet of a dread with this ability.
However, there's precedent for the idea that negative effects don't usually hit the person who cast them even if they're in the center of the effect. Most other "Aura" abilities specify; for example, Elemental Aura:
Creatures adjacent to you when this spell is cast and at the start of your turn take 2d6 points of energy damage of the selected type.
(you cannot be adjacent to yourself)
Any hostile creature within a 20-foot radius of you must make a Will save to resist the effects of this aura.
(you are not hostile to yourself)
Couple that with the preceding sentence which specifically calls out enemies as being effected by the aura and you have what I believe to be the intent: the aura only effects enemies, period. You and your allies do not lose immunity to fear.
Dreamscarred Press Specifics
It was pointed out in the comments that my examples are core Pathfinder material from Paizo while this class was written by Dreamscarred Press. So I combed the Psionics material and found more examples of this kind of confusion. Corrosive Aura:
You surround yourself with a corrosive aura, damaging all who come near. Each creature that ends its turn in your square or adjacent to you takes 1d6 points of acid damage.
By definition you always end your turn in your square. So you'd take 1d6 damage a round.
Worse, Regenerative Aura clearly intends to include you:
Psionic power radiates out from your body, causing all living creatures within range to rapidly heal as their healing ability is hyperaccelerated.
Untouchable Aura is the only one written to exclude you:
Anyone trying to make a melee attack against you must succeed on a Will save or lose their attack action as they recoil in fear from the images inside their mind.
(Unless you attack yourself).
If I were the DM, I'd just rule that DSP is terrible at explaining what they mean and go with the solution I offered earlier, because this stuff is giving me a headache, but there's evidence to support a ruling the opposite way on both this and Corrosive Aura. I'd not rule one of them one way and the other the other way, I'd go both or none.