I'm Lucy, and I'd be happy to answer your questions today. I'm sorry to hear that this happened.
Parking with an expired disabled placard is a strict liability offense - either you were or you weren't. The law doesn't require that you meant to do it, or say that anyone who does this is a bad person or anything like that. The expiration date is printed on the front of the placard, so it's unfortunately not a defense that you weren't aware it had expired. It's up to the person who wants to use the placard to ensure that it's valid before parking in an accessible space (even if it weren't printed on the front, this would be considered your father's mistake, not the DMV's. An understandable mistake, but still his mistake). Legally, you could actually sue your father for that money, since he's the one who caused you to use the expired placard. But I can, of course, see why you probably don't want to do that.
If you received paperwork saying that you needed to pay the fine in order to file the dispute, and you didn't, then it's unlikely that you'd be able to reopen the case. It sounds like what happened is that you requested a Trial by Declaration, which means a person sends in the fine and a statement of what happens, and asks the court to find that they're not guilty and return the money. If something like this happens again, you should know that the way to avoid this problem is to go to court and talk to the DA before the trial date. They'll usually work with you on making payment arrangements or lowering the fine if they can.
Call the clerk of the court to ask if they will put the case back on the calendar, since you missed the court date. You want to make sure that you didn't get charged with a failure to appear as a result of this mistake. They may be able to reopen the case for you if you are able to go to court and arrange for you to do the fine in installments or do community service to pay it off, if you can.
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