They were swimming with the sissies!

Reputed mobster “Tony the Wig” cowered under the protection of a pretty parasol outside Manhattan federal court Friday as his Genovese associate John “Tugboat” Tognino fumbled with his feeble umbrella under a gust of wicked wind.

The wiseguys pleaded guilty to running illegal poker tournaments, dice tournaments and taking bets on horse races.

Anthony “Tony the Wig” Vazzano and “Tugboat” Tognino copped the pleas as part of a sweeping deal with the feds that will have dozens of accused mafiosos pleading guilty in the next few weeks.

A total of 37 accused mobsters are expected to plead guilty by May 19 as part of a deal hashed out with prosecutors last month — including Arthur Avenue restaurateur and reputed Genovese capo Pasquale “Patsy” Parrello, sources familiar with the case told The Post.

The case began last year when the feds rounded up 46 men from four of the five major crime families and charged them with working together on illegal schemes including gambling, credit-card fraud and health-care fraud.

But the feds’ case started to show cracks in March when Manhattan federal prosecutors warned the judge that two FBI agents were being probed for interactions with a key witness.

In a sealed letter, prosecutors said they were investigating a failure by the agents to archive debriefings with cooperating witness John Rubio, a Genovese associate who wore a wire and secretly taped his pals.

By April, the feds agreed to drop the racketeering charges, which carry up to 20 years in the slammer — but only if enough people agreed to plead guilty.

Under Friday’s agreed-upon deal, Tognino, 75, faces between four and 12 months in prison.

The mobster told the judge he suffers from a gambling addiction and claimed he was treated for “pathological gambling” in 1998. That same year he was indicted on money-laundering charges in Las Vegas in a sports-wagering case. The charges eventually were dismissed.

Vazzano faces between six and 14 months.

Tognino and Vazzano both told Judge Richard Sullivan that they worked collecting money in New York tied to bets made through an online gambling organization based in Costa Rica.

“I did go and collect winnings,” said Tognino, who has seven convictions over 38 years, mostly tied to gambling.

He said he only collected bets from “friends of mine,” but admitted he knew it was against the law.