The Spain international debuted his new haircut in north London and Moreno conceded a first-half penalty following a foul on Arsenal forward Theo Walcott.
While the England striker missed his spot-kick attempt, Moreno was caught out of position minutes later as Walcott broke the deadlock with a low finish.
Liverpool went on to secure three points thanks to goals from Brazilian Philippe Coutinho, England star Adam Lallana and new signing Sadio Mane.
Moreno has received plenty of criticism from former professionals after his disappointing performance at the Emirates Stadium on Sunday.
And former United right-back Parker believes Jurgen Klopp will look to sign a left-back before the close of the summer transfer window.
“It’s such a shame about Joe Gomez who would do very well under Klopp but I don’t know how far away he is from fitness,” said Parker, speaking to 888sport.
“I think Klopp knows he now needs a left-back and if I was Moreno I wouldn’t have my hair in a stupid ponytail. I wouldn’t attract that much attention.”
The Reds are in second place in the Premier League table behind Manchester United on goal difference.
The 18-time English champions will travel to promoted side Burnley in their next Premier League fixture on Saturday afternoon.
Read the full 888sport interview with Paul Parker, including his thoughts on Clyne being the best England right-back option
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