Here is a draft representation to lay the groundword:
Dear London Borough of Lambeth,
On the day of the alleged contravention I was accompanying my daughter to Kings College hospital, I was taking her to the Orthopedic & Trauma department for a scheduled appointment at 3pm. I duly paid for my parking session but as matters were taking longer than expected, attempted to extend the parking session but I had no reception. I was not in a position to leave my daughter unattended and despite repeated attempts, I was not able to extend the parkin session.
I hope that in light of these mitigating circumstances you will agree to cancel the penalty charge.
Yours faithfully,
Do not supply any supporting evidence at this time, we want to keep that for the notice to owner stage.