The Who also took to Twitter to offer their "condolences to the family and friends" of the British guitarist.
Rock star Peter Frampton and Parfitt's friend, the broadcaster Chris Tarrant, were among the others paying respects.
Parfitt died in hospital in Spain, from a severe infection following complications to a shoulder injury.
His partnership with Francis Rossi spanned five decades, making Status Quo one of British rock's most enduring acts.
'He was a mechanic and I was the receptionist': Your memories of Rick Parfitt
Parfitt had suffered a series of health problems including having to undergo a quadruple heart bypass in 1997. In October, he said he would no longer be performing with his band after suffering from a heart attack in the summer.
But he had been due to launch a solo career with an album and autobiography planned for 2017.
In a statement on the band's website, his manager Simon Porter said Parfitt been admitted to hospital on Thursday.
His family was "truly devastated" to announce he had passed away in Marbella at lunchtime on Saturday, he said.
"He will be sorely missed by his family, friends, fellow band members, management, crew and his dedicated legion of fans from throughout the world, gained through 50 years of monumental success with Status Quo."
In a later update, Porter thanked people for their messages of sympathy and support.
"Rick had come to terms that he would be unable to tour with Quo again, but he continued to be very much part of the Quo machine and we discussed future plans for the band," he said.
He added: "I really think that we all thought that Rick was invincible, given his numerous and astounding recoveries from situations which would certainly have killed the average human being.
"Indeed, early this morning when I initially heard the news that he had taken a serious turn, my mind instantly thought that he would come around by Boxing Day...
"Quo fans should know and take comfort in the fact that his passing has, ironically, come at a time when Rick was in a happier place career wise than he had been for a long, long, time."
Parfitt is survived by his wife Lyndsay, their twins Tommy and Lily, and his adult children Rick Jnr and Harry, a statement from the family said.
Rick Jnr tweeted: "I cannot describe the sadness I feel right now. To many he was a rock star, to me he was simply 'Dad', and I loved him hugely. RIP Pappa Parf."
Drummer Jeff Rich, who played in Status Quo with Parfitt for 16 years, told the BBC he was "just a really good musician, very under-rated, great rhythm player", but his health problems were "a big issue for him".
He added: "When he had the last heart attack and he almost died a few months ago, I had a feeling that he wouldn't survive very long but in the back of your mind there's always this thing thinking maybe he's going to be alright now - but when I got the phone call, you know I was really upset."
Broadcaster Chris Tarrant, who knew Parfitt for more than 40 years, said: "I am truly shattered to have this news. We all honestly felt that he was getting strong again.
"Rick was the ultimate rock and roller. He lived and died as he wanted to... the world is a sadder and poorer place without him".
Status Quo found global success with hits including Rockin' All Over The World and Whatever You Want.
Queen guitarist Brian May tweeted: "Shocked and so sad to hear of the passing of Rick Parfitt. Hard to find words, You truly joyfully rocked our world. RIP dear buddy."
Peter Frampton said he was "so sad" to hear the news.
Meanwhile, Former Ultravox frontman Midge Ure wrote on Twitter: "Dreadfully sad. Lovely man. Thoughts go out to his family and friends."
Ure co-organised Live Aid in 1985, which was memorably opened by the Quo performing Rockin' All Over The World.
By Nick Serpell, BBC obituary editor
With his flowing blond locks, denim gear and Fender Telecaster, Rick Parfitt was one of rock's most recognisable guitarists.
His partnership with Francis Rossi became the core of Status Quo, one of Britain's most enduring bands.
Their brand of boogie-woogie rock survived changes in musical fashion and made them one of the best-loved live acts of their generation.
As well as driving the Quo sound on stage, Parfitt penned many of the band's biggest hits.
Read more about Parfitt here
Many other musicians paid tribute on Twitter.
Chas and Dave, who toured with Status Quo in 2014, tweeted: "Very shocked and saddened to hear of the passing of our friend Rick Parfitt".
Rock band Uriah Heep tweeted: "Such devastating news that Rick Parfitt has passed away.  Rick left a musical legacy that has given such joy to everyone around the world."
Def Leppard frontman Joe Elliott wrote: "R.I.P. @Status_Quo's Rick Parfitt - We grew up on Status Quo as kids & have a great respect for them & their legacy.. - Joe".
Novelist Ian Rankin wrote that he would be "playing air guitar" in memory of Parfitt.
In 2015, the band passed a milestone achieved by only a handful of musicians, spending a total of 500 weeks in the UK album charts.
Status Quo superfan and secretary of the band's official fan club Yvonne Hanvey told the BBC: "It's just absolutely devastating.
"We knew it would happen at some point. It does with everybody obviously, but Rick was almost invincible.
"He believed he was invincible. And this is just absolutely unbelievable."
Queen guitarist Brian May has paid tribute to Status Quo star Rick Parfitt following his death aged 68, saying he had "truly joyfully rocked our world".