It's a pity you didn't take a photo, but realistically, I suppose that is not 'proof' as anyone could move a car.

I'd simply write it up as clearly as possible. State exactly where you were parked-street name and even near what numbers, the exact spot and times you were parked and date. Explain there are no parking restrictions in that street.

Then point out the discrepancies and the fact the street named on the ticket is 3km away.

Do not expect the council employee who reads it to know automatically that the streets are not near each other or that your street has no restrictions. You will have to spell it out to them. Good luck.