I don’t understand your letting agents.
If a property doesn’t have off road parking it is valued as such and that rental figure used. Unless it is one of very few in the whole area which doesn’t have off road parking, so no other benchmark, there is no basis for deducting a percentage.
I have no idea if the letting agents promised off road parking to the tenant. From the tenant’s communication it’s seems likely they did, but it makes no sense. They made a big point of valuing the rent without off road parking. It’s only a one year tenancy, and the work would involve huge expense, so why would they?
The landlord’s contact details are normally shown on the Tenancy Agreement, and that is probably how the tenant got them. The tenant seems to be under the impression that the landlord is at fault.
OP you need to look at the written contract to make sure that nothing has been included about off road parking.
Either way this is a difficult situation. The letting agent is YOUR agent, and if they have made inappropriate promises it is not the tenant’s fault. They are entitled to rely on promises made by your agent.
It could be the case that the tenant has some claim for damages against you as landlord, which you would then need to pursue from the letting agent.
Realistically it’s unlikely to get to that stage and they probably don’t have much of a case for financial compensation, but they are already indicating withholding rent, which makes it a problem.
The very first step is to notify the tenant and letting agent, in writing, that you categorically did not and do not agree to provide off road parking, and nor do you give consent for this work be done by the tenant. You also need to tell the letting agents very firmly that they must reiterate this to the tenant, and deal with any consequences.
You are possibly at the stage where you could issue a Section 21 notice for them to vacate at the end of the fixed period in July, but I think the notices are only valid for 6 months, so best check. I wouldn’t rely on your agent.
I don’t think the tenant is right about being able to get the work done themselves and charge/deduct from rent. There is some provision for this for essential repairs and maintenance, but it seems unlikely that an improvement of parking facilities would count.