Yes by all means start drafting something up and show us a draft for comment.
Your main points will of course be around the rights granted to you in your lease, your rights over the space in question, the lack of contract between you and P4 (you already have a right to park in the space as part of your tenancy, a right being exercised at the time, and therefore did not enter into a contract with P4 whereby you agreed to pay £100 to park).
There's a fair chance POPLA will reject you, not because your arguments are without merit, but because if they accepted your arguments, it would ruin the operating model of most of these firms on residential land, who make much of their money penalising innocent mistakes by residents.