He has now agreed a deal with the Serie A side after his Blues' contract expired.
"I thought for a couple of hours after [the play-offs at] Wembley that I had a great opportunity to stay in Swansea and play in the Premier League, which I would like to do," he told the Swansea website.
"Maybe it's a missed opportunity not playing in the Premier League, but maybe this is a good thing for my career and will make me stronger.
"I have no regrets, Parma is quite a big club with a big programme for the future. Maybe I will do well there and I can come back to the Premier League later in my career."