Air carrier flight crew reported uncommanded aircraft movement due to failure to set parking brake after push back. Flight crew set the brakes.

Narrative

Early morning departure after min rest night before. Dealing with distraction of possible de-ice due to cold weather; Arrived at airplane to maintenance issue involving DMC (Display Management Computer) and Captain screens inop. Resolved issue with Maintenance Control. Followed by issues with performance and heavy aircraft errors causing distractions. Hot mic on Ramp mic caused even more distractions as they were calling for push and brake release with cargo door open. Finally completed Before push and started push back. After brake release and in push we received company message regarding lavatory service completion being in question. Told ramp to stay connected. This caused another distraction and we never received set parking brake call. We contacted FAs to verify lavatory and contacted Operations to have them clear the message. Due to the distraction parking brake was not set. Ramp disconnected tug and left per SOP. When we started #2 engine I could feel airplane move forward and stopped it rolling. Set the parking brake and took time to evaluate situation. Airplane was never in jeopardy; but had we not caught it; it could have been. We did not feel rushed or time pressure but the break in push back; sop of hot mic; and nonstandard calls combined with multiple distractions caused this issue.