The fixture, originally due to take place in early March, was rescheduled amid Parma's financial woe, with goals from Iago Falque and Leonardo Pavoletti lifting Genoa up to seventh in the table.
Parma had enjoyed rare success in the last week, with a 1-0 win over Udinese followed up by a shock victory by the same margin over runaway leaders Juventus on Saturday.
However, there was to be no third straight triumph as Falque profited from Marco Borriello's incisive run down the right to slot home in the 14th minute.
Pavoletti added a second 14 minutes from time after a perfectly weighted pass from Diego Perotti as Genoa climbed to within five points of the UEFA Europa League spots, while Parma sank ever closer to seemingly inevitable relegation.
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