I think, technically, yes, but....
A PC should receive a predetermined level of parking and traffic enforcement cover from the LA. It should be part of a SLA.
If the service delivered is considered to not meet the needs of the community - whether that be for a specified event / timeline or on a more permanent basis - then either a PC or a group of PCs can apply to LA for additional parking / traffic enforcement cover from LA dept at an additional cost. Cost and hours can be 'shared' across more than 1 PC area.
A PC may consider resourcing their own civil enforcement officer but the cost benefit for recruitment, training, PAYE, leave and overly generous civil service terms and conditions rarely makes sense for what is probably a few hours work per week.
First port of call work be to touch base with LA parking enforcement and establish what are the established parking enforcement arrangements (if any) and what might be possible if established levels are inadequate.
If that doesn't deliver results then somebody has to quantify the extent of the "problem" and calculate the cost of direct employed staff to remediate.
Then it is a case of "problem" divided by "cost" equals the decision whether it is credible or not.