It seems highly unlikely that a single parking ticket could possibly result in a $1,300 penalty! More than likely, there are a number of outstanding parking tickets. Has a judgment been entered against you for $1,300, or is the matter still open? If it is open, the agency involved is likely to negotiate a settlement with you, since it is almost 20 years old and they would probably want to "clear" their records os old cases. Also, it is common that a payment schedule could be worked out, given the amount and the economy.
Finally, if it is a parking ticket, the vehicle owner is responsible even if you were "somewhere else" at the time.