As ever when such questions arise, opinion seems to dominate regulation.
The regulation is as set out below:
https://www.legislation.gov.uk/uksi/2003/1021/part/5/made
26(1)(a) appears to 'allow' for the payment of travelling expenses to attend a council meeting by bicycle or any other non motorised form of transport. A little antiquated perhaps...
Para 25 - Parish basic allowance - is a much more controversial topic and one which appears to be vehemently opposed in certain quarters - I would submit that perhaps some of the difficulties associated with recruiting and retaining decent Cllrs, especially from a more youthful demographic, might be alleviated by an objective examination and discussion of this topic.
The suggestion that Cllrs voluntarily undertook the role and so should cover their own costs and that walking to a council meeting might be appropriate are archaic and poorly considered but probably widely held.
Walking to a council meeting in, for example, a rural parish with an area of 4500 hectares and with a distance of 7 miles between the North and South, 6 miles between the East and West boundaries, the majority of which is open and exposed, unlit and unpaved upland moor, in Winter, by often elderly and/or infirm individuals (probably unaccompanied) is patently ridiculous and counter to the contradictory duty to consider 'inclusivity' and accessibility.
Maybe the reference to the 'ability' to pay for bicycle rates will assist the immediate discussion on travel expenses if only to serve as the 'reason' why motor charges might not be lawful. Perhaps an amendment to the motion to consider Cllr allowances more generally would be a better approach but that is for each council to consider...