The group had been due to play on Saturday night in Pula, Croatia, where Parfitt, 65, is now in hospital receiving treatment.
No details have been given, but band manager Simon Porter said Status Quo did not "cancel shows lightly".
"In this instance we have no option but to follow medical advice," he said.
Parfitt had a quadruple heart bypass in 1997 after doctors said he was in danger of dying as a result of his lifestyle.
After surgery, Parfitt said he was not planning on becoming a "born-again Christian" and would still have the "odd pint".
His latest illness has forced the band to cancel dates in Monte Carlo, Switzerland and Germany up to 10 August.
There was no immediate word on whether Quo's UK tour dates later this month and in December would be affected.
The band have recorded 64 British hit singles since 1968, more than any other band.
Their hits have included Down, Down, Rockin' All Over the World, Whatever You Want and In The Army Now.
Parfitt and long-time band partner, Francis Rossi, were invested as OBEs by the Queen in 2010.
2 August: Pula, Croatia
4 August: Monte Carlo
6 August: Schaffhausen, Switzerland
7 August: Biel, Switzerland
9 August: Tettnang, Germany
10 August: Dautphetal, Germany
Rock band Status Quo have been forced to cancel six concerts on their European tour because of illness affecting guitarist Rick Parfitt.