"I love my wife," Parker said in a statement to PEOPLE. "She's the best thing in my life, and I have never been happier." Longoria, who took Parker's name this year after their July nuptials, added, "Tony has been nothing short of the perfect husband."
Earlier, Longoria's rep Liza Anderson had called the allegations "completely, 100 percent false and untrue."
JOSH & FERGIE: ROCKED BY SCANDAL
Did he cheat with a stripper?
Married less than a year, the couple denies an Atlanta woman’s claims that she and Josh had a fling
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