When parking where you're going to day hike, no part of the vehicle should touch a road or the natural area. Other drivers need to get through, and you wouldn’t want a wide load or a careless driver to hit your vehicle.
You also don’t want to needlessly damage the very area that you plan to hike, so don’t park on the grass or in what appear to be “weeds.” In some places, such parking can net you a ticket.
You may need to pay for parking. Usually it’s included in any entry fee or pass to a park. In many places, day use for hiking is free so long as you aren’t camping. National forests typically require a pass for day activities; you can buy them either annually (the best deal) or for daily use. They are available at most outdoors and sporting goods stores near the national forest.
Read more about day hiking with children in my guidebook Hikes with Tykes.