For a TFL penalty, just replace the word "council" with Transport for London, as that is the authority you have to deal with.
There is no right of appeal against a PCN, there is only a right to make representations to the enforcement authority (which in this case is Transport for London). You can make representations on whatever grounds you want, some grounds are statutory grounds of appeal (these are listed on the PCN itself), anything that isn't a statutory ground of appeal is known as mitigation.
Transport for London has a power to accept representations on whatever grounds it wants, whether they are statutory grounds of appeal or not.
If the representations are rejected you have the right to appeal the London Tribunals, but the tribunal can only allow an appeal based on the statutory grounds, the tribunal adjudicator cannot allow an appeal on non-statutory grounds no matter how compelling the reasons might be. But if Transport for London fails in its duty to consider the representations, that in itself gives you a statutory ground on the basis of which you can appeal to the tribunal.
I suggest you draft up something based on what you've told us re: your later father's birthday, don't worry about setting up a link with a click counter as I will do that for you.
Do you have the death certificate to hand?