Hi TS, I think this is quite common problem/issue in housing area...at least that is what I think. When I move to a new place (few years ago - landed housing area), I decided to 'make a good relation" with the neighbours...
This is partly got to do with...possible problem that might occur in future...like, parking (like in your case), too much noises, etc...

What I trying to say is...is a 'give and take' situation....
I not sure is it too late for you to try to make the relationship better...
Personally if I am your neighbour...I will try to understand your situation (means let you park for a while...maybe that day you got some friends visits or etc).
I will only hate it when...it is obvious someone just dont like to park in front of their gate (for convenient for them to drive in-out of their other car that parked inside the gate)...everyday.

I'm not saying who is right or wrong here...just that, I feel that, we cant assume others people will think the same as we/me...especially like in your case, some feel is all right, some feel...is not all right...
That is the 1st thing. On the second thing, try to talk to this old lady...the earlier the better...(although I feel...quite late for it...but heck, worth to try).