Deathbringer Regent and Phantasmal Image ETB Clause
Yes, it will count as having been cast from your hand.
It's important to note that a creature spell and the creature it eventually creates are different objects; putting a creature on the battlefield is the effect of a creature spell. If you cast a creature spell from your hand, and if an effect changes the characteristics of that creature as it enters the battlefield, the creature spell was still cast from the hand.

601.2. To cast a spell is to take it from where it is (usually the hand), put it on the stack, and pay its costs, so that it will eventually resolve and have its effect.
608.3. If the object that’s resolving is a permanent spell, its resolution involves a single step (unless it’s an Aura). The spell card becomes a permanent and is put onto the battlefield under the control of the spell’s controller.

Phantasmal Image's effect is a replacement effect.

614.1c Effects that read “[This permanent] enters the battlefield with . . . ,” “As [this permanent] enters the battlefield . . . ,” or “[This permanent] enters the battlefield as . . . “ are replacement effects.

In your case, it replaces the effect &quot;put a Phantasmal Image creature on the battlefield&quot; with &quot;put a copy of a particular Deathbringer Regent creature on the battlefield&quot;. All other things remain the same, in particular that the spell was originally cast from the hand.