The band's manager Simon Porter added: 'No further comment will be made at this stage until the completion of the next round of tests and assessments to be made over the next few days.
'We would ask you to respect the family's privacy at this difficult time.'
A self-confessed former drug addict, Parfitt has suffered a catalogue of health problems, including a quadruple heart bypass in 1997, a throat cancer scare in 2005 and a heart attack in 2011.