Burnley lost 1-0 at Turf Moor thanks to an injury-time penalty converted by Mohamed Salah.
Parker said afterwards, “My overriding feeling is one of pride. We were brilliant today against an incredible side.
“Of course I’m gutted, absolutely gutted. Football can be a cruel game at times - and we’ve learnt that the hard way in our last two Premier League fixtures, but I’m immensely proud of the group.”
On Hannibal Mejbri's hand-ball call for Liverpool's penalty, Parker added: “I don’t really want to discuss the penalty to be honest.
“The ball came at him at possibly 70-80mph from three yards away, so I can’t fault Hanni at all.
“I think my biggest takeaway of that decision is that maybe we’ve got to look at coaching players to block balls with their arms behind their backs now. That’s new to us all but it’s where the game is.”