Rodriguez, who has played for Uruguay at two Copa America tournaments and the 2014 World Cup, joined from Atletico Madrid, where he has made only six league appearances this season, all as a substitute.
The former Paris St Germain, Benfica and Porto player has not previously played in Serie A.
Varela, who like Rodriguez is 29, has been loaned by Porto, having spent the first half of the season at English Premier League West Bromwich Albion, where he scored one goal in seven league appearances.
Parma have mustered only nine points in 19 games so far, with three wins, one draw and 15 defeats, and are nine adrift of safety. They were deducted one point in December for falling foul of Serie A financial regulations.
Roberto Donadoni's team, who finished sixth in Serie A last season, also missed out on the Europa League for not meeting UEFA's club licensing criteria.
(Writing by Brian Homewood in Berne; editing by Toby Davis)
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