"@JoeBiden, please get this done,” tweeted Ms Clinton.
The award is given by the president to an individual who has made exceptional contributions to the country.
In addition to her iconic country music career, Ms Parton donated $1m to help develop a Covid vaccine and started a foundation that has donated more than 100 million books to children.
Mr Obama admitted his “major” mistake during an appearance on The Late Show with Stephen Colbert as he promoted his book, A Promised Land.
“How does Dolly Parton not have a Presidential Medal of Freedom?" he was asked by the late night talk show host.
"That's a mistake, I’m shocked," said Mr Obama.
“That was a screw-up. I'm surprised. I think I assumed that she had already got one, and that was incorrect.”
The former president added that he will try and fix the situation.
"She deserves one. I'll call Biden," he said.