I'm not so sure, so I contacted the BBC (as Mr Parris is a long-serving Radio 4 presenter) to ask what action hey would take.
You can contact the BBC (Radio 4) directly using their BBC Programmes Feedback Form. Here is what I asked:
Will you be considering Matthew Parris' position on various Radio 4 programmes, following his outspoken and inciteful remarks regarding cyclists?
19 April 2019 (Friday): Gt Yarmouth CC Good Friday 10-mile time trial
11 May 2019 (Saturday): VC Norwich 25-mile time trial