Borders, Falkirk and Stirling Council had considered suing after running out of time to replace traffic wardens, which were previously run by police forces ion their areas.
However, the threat has been lifted after more time was given to come up with a way round the dispute.
The row was sparked after Police Scotland announced in October last year that traffic wardens were to be phased out to save money and standardise services.
Traffic warden services in Scotland have been run by police in some local authority areas, while others took control decades ago.
Police Scotland wished to see all services operating at the same level, but this was opposed by the three councils and will now be postponed for two months.
Stirling Council's Public Safety Committee Convenor Councillor Martin Earl said: "The way Police Scotland has gone about this has shown scant regard for proper consultation with local councils or the impact of withdrawal of traffic wardens on the public.
"It is surely wrong for Police Scotland to pick and choose which laws it will enforce.
Assistant Chief Constable Wayne Mawson said: "Discussions are ongoing with local authorities who have corresponded with us over the change of traffic warden provision.
"In those areas where traffic warden service has been withdrawn, police officers have continued to work with local stakeholders to identify areas affected by problematic parking, targeting them during periods of directed action."