Yaroslav Zheleznyak, an opposition member, announced the decision on the Telegram messaging app, saying 313 deputies had voted in favour in the 423-member Verkhovna Rada assembly.
Zelenskiy had vetoed an earlier version of the bill and called for the two symbols to be allowed in displays in museums, libraries, scientific works, re-enactments, textbooks and similar instances.
Neither of the two letters exists in the Russian alphabet. They have been widely used, particularly on Russian military vehicles and equipment, to promote the aims of the conflict.
Moscow calls its invasion of Ukraine a "special military operation" to disarm its neighbour and protect it from fascists. Ukraine and the West say the fascist allegation is baseless and the war is an unprovoked act of aggression.
Over the weekend, Russia pummelled positions in the east of Ukraine, pounding the Donbas and Mykolaiv regions with air strikes and artillery fire.
The new bill bans the creation of non-governmental organisations using Russian war symbols or undermining Ukraine's sovereignty.
The Ukrainian parliament on Sunday also extended for another 90 days, or until Aug. 23, the period of martial law in the country.
(Reporting and writing by Ronald Popeski in Winnipeg; Editing by Chris Reese)