Only $20 for 6 hours of care and transportation?! Um, 'No as my day is done at X time." If you want to work later, tell her, "You can pick up child an hour and a half later if you want to pay my after hours rate: $20 per hour, so that would be an extra $30" You leave the ball in her court and she probably won't ask again.
Your daily rate is very low, are you licensed? You refer to the rest of us as "babysitters." Many of us are licensed professionals and our rates are at least double or more than yours. If you want to stay in this business, consider getting licensed and upgrade your skills and then you can see yourself as a professional with a valuable service to offer with a fee that reflects that fact.
The parent is treating you as cheap labor. Don't allow it.
