To date two or more girls simultaneously. This refers to the 1/19/62 Flintstones episode #44 "The Entertainer" when Fred has to manage two dates at the same club when Wilma unexpectedly shows up. Of course, he had the help of Barney to distract them as he switches between girls (this will not always be your case). Even though he was ultimately unsuccessful, the expression remains.
Man, I forgot about Caroline and invited Audrey out too... I can't cancel now, I'll have to pull a Fred Flintstone.