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 January 2020 (Sunday): Iceni Velo First Self Drive Cross
26 January 2020 (Sunday): Milton Cross