OP, from the lease's legal terminology and format..
with the residential use and occupation of the property and the Parking Space
The right to park one roadworthy taxed and MOT'd vehicle not exceeding two and one half tonnes gross unladen weight in the Parking Space
From your freehand intro..
The only areas where parking (even then, not directly, just a reference to a parking space)
Spot the difference.
The lease refers to 'Parking Space' whereas you refer to 'parking space'.
The capitalisation of the 'P' and 'S' must surely mean that it's a defined term in the lease. So, please look for this, maybe in a Schedule, Annex or plan of the site, in each case this would be referred to in the main body of the lease.
Your lease is pretty standard:
What you hold as part of the property leased by you (I suggest this includes Parking Space);
Your general rights and obligations as regard use of the other, common, parts of the site;
Your general obligations to the landlord in order to maintain good order within the site.
It would also likely make reference to how amendments to the agreement (a lease is a form of contract/agreement) may be effected and to how disputes and breaches are to be resolved and we should see this as well.
My take on these is that the landlord has contracted with a company (property management) to discharge their responsibilities to you under the lease and to manage and oversee the general maintenance and functioning of the site. Such a company is the agent of the landlord, not independent, their authority is circumscribed by the terms of each lease and your obligations remain unaltered other than as provided for under the lease. Use of a private parking contractor is often sensible in order to protect your and other residents' rights. The problems come when a naive property company contracts with a money-grubbing contractor(look, no fee!) without thinking of the potential ramifications.
Your use of a permit is an administrative measure by which those who are not permitted to park may be identified and thereby assist the PPC and mgmt company discharge their obligations. It cannot alter the terms of your lease.
Anyway, pl provide the missing references within your lease to Parking Space.