The comedian told how their tradition started after he had forgotten to send the Beatles legend a card one year.
The Big Yin had asked his daughter to arrange a card but he discovered it had not been posted three months later and his wife Pamela Stephenson sent it.
Connolly, 73, revealed the pair now try to outdo each other by sending festive greetings as late as possible each year.
In an interview in the US, he said: “I have got a funny thing going just now with Paul McCartney.
“My daughter was in charge of sending all the Christmas cards away and she hid them in a cupboard and did something else and forgot about them.
“Then my wife came back in February and discovered them and sent them all.
“I got a lovely letter from Paul McCartney saying ‘Thanks for the Christmas card, it’s made March such a nice month’.
“And so I got one from him this year in March. So I sent his three days ago.”
Connolly said despite meeting famous names including The Beatles, The Rolling Stones and The Queen, the only person who has made him starstruck is Bob Dylan.
He said: “It’s funny, I have met Bob Dylan about three times and he never knows me.
“The last time I met him we did the David Letterman show together. We were talking about songs and stuff but he was staring at me and I could tell he didn’t know me.
“I have got a picture of the two of us at home, it’s from Blackbushe airfield near London where they were having a big concert with Eric Clapton and him.
“I was backstage and he came up and was talking to me but I wasn’t saying anything back because he was behaving as if he knew me.
“I thought if I say something he will hear my accent and he will know he doesn’t know me and he might stop. So I was just listening to him.”
Last week, Connolly admitted he has had suicidal thoughts since being diagnosed with Parkinson’s disease.
He was diagnosed with Parkinson’s in 2013 just weeks before he had successful surgery for prostate cancer.
The Glaswegian said he now struggles to get out of bed some days because he is in so much pain and is now unable to play his beloved banjo.
He has taken to meditating to relieve bouts of depression and said he is still hopeful a cure can be found for the illness.
Asked whether he had considered “checking out” on his own terms, he said: “Yeah sometimes I give it a bit of thought when I’m in bed.
“I think ‘Well this is forever, this isn’t going to get better, it’s going to get worse’.
“But then I try and change my mind and I try and meditate and move away from it sideways.
“The guy who told me I had it said to me ‘You realise it’s incurable?’ I thought he could have said ‘We have yet to find a cure’ or something like that to put a bit of light at the end of the tunnel.”
He added: “I’m okay at the moment but it comes and goes. Sometimes I have trouble getting out of bed and I walk sort of strangely.
“Turning over (in bed) is difficult. Turning from one side to the other can be quite a complicated manoeuvre.
“The body isn’t responding sometimes but it is quite interesting as the body changes.
“I have started to drool as well, that’s a nice thing. That’s going to make me really attractive.”
Despite his health troubles, Connolly is set to embark on a three live stand-up shows in America.
He plays the Beacon Theatre in New York on Wednesday followed by dates in Washington DC and Boston.