Air carrier pilot reported after pushback the aircraft began to roll during engine start. Pilot cited a communication breakdown and possibly not having set the parking brake as contributing to the event.

Narrative

Issue with no parking brake at engine start...and tug not attached. Pushed out of ramp X; spot XX. Tugged all the way back to the center line...even thought that we were really going for a long push back. As we stopped moving backwards; mechanic stated cleared for start. (While I would like to say that I am 100 % sure he did not say parking brake set; there is just the smallest chance I may have missed something. but I am pretty sure I did not say parking brake set to his call for it; as I would have set the brake). With that; it is normal to start the engines while still hooked to the tow bar/tug. Just because you are cleared to start does not mean you set the brake. So number 2 started normal; Cleared to start number 1. As number two spooled up; with both myself and the FO monitoring the gauges for the start; the mechanic excitedly called us and told us we were moving; twice very rapidly. brakes were applied to stop very slow roll; then parking brake set. Roll was very short and slow as we did not feel it. No issues for remainder of start. I cleared off the mechanic and taxied normally; a little flustered; but normal.We talked about how this could happen; did I not set the brake? Did I not set it all the way? Why would I not set the brake? Did he call for the brake to be set? Like I said; I may have been at fault; and I take full responsibility if so; but I feel that since I did not get a parking brake set call from the mechanic; the brake issue was out of the normal habit pattern. Hence; we had no brake set after the tug left the aircraft during start. I am not trying to point fingers; hopefully not at myself; but this was one of those cases where a call was missed and the Swiss cheese lined up. We as a crew talked about how to handle this type of situation in the future. In the future; we are going to try and be a little more aware of our out side situation after a cleared for start while still hooked to the tug. It's interesting how easy something like a missed call can get you out of your normal habit pattern; and while this was an attention getting moment; we were lucky that we were not near anything and that the mechanic noticed the slow roll.