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?
28 January 2018 (Sunday): Cambridge Cross
11 February 2018 (Sunday): Ely & District CC Hardriders 25-mile time trial