Is it possible to just pull over somewhere and ‘camp’ not in a park?

The answer to this is:

Yes.

No.

Maybe.

In the USA, there are strong state laws which apply to individual states. Some laws for different states are similar and some vary widely from state to state.  So part of the answer is it depends upon just *where* you are!

Camping overnight not in a designated campground or RV park is called "boondocking" or sometimes "dry camping".  The latter term also refers to camping within a campground, but with no hookups.

Note:  Tenting is almost never allowed anywhere except in a designated campground, or in the back country with a camping permit. Never trespass on private property to tent camp.

--> At any time an RVer wants to park overnight not in a campground, they must obey any signs that say "No Parking" ... "No Overnight Parking" ... "No Camping" or similar. Do not camp anywhere on private property without the owner's permission.

-->You cannot camp overnight in Day-Use State or County Parks. 

-->Within the boundaries of a National Park, you *must* park overnight in a designated campground, in an assigned space. Many National Parks require campground reservations well ahead of time; some have “first-come-first-served” campgrounds – many will have both.

National Forests do allow what is called "dispersed camping". There are certain rules and restrictions that must be followed if you decide to do this, and they may vary in different forest areas. Be sure to check online for regulations in the specific National Forest you will be visiting.

Note the difference between the National Park boundaries and the National Forest areas!

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Places where you can park an RV overnight:

--> In general, you can park overnight for free at truck stops. These also have food, fuel, a small store, toilets and showers($).  Please park out of the way of the big rigs!  It is a good idea to check in with the store clerk or office.

--> Many Indian casinos allow overnight camping for free. You may have to register, and you'll need to park in an assigned RV lot. Many casinos have restaurants. 

--> Rest stops are located along freeways and highways. At some, you are allowed to stay overnight in an RV. At some, you are not. Some are not in particularly safe areas such as near a big city. Rural rest stops are overall safer, but be aware there is no security. If a rest stop is signposted "No Camping", do not park overnight in your RV.

--> Walmarts. Some Walmart stores allow overnighting, some do not. Many cities and towns have local ordinances preventing RVs from parking overnight in a Walmart or other large parking lot. Walmarts in rural areas and smaller towns tend to be more lenient about this. It is best to enquire inside the store before setting up for the night. 

--> As far as camping ‘alongside’ the road, there is a wide variety of rules and regulations on this! Some states allow it, some do not. Some areas - even within states that permit it - will not allow it for various reasons.

You should NEVER just park on the shoulder of a road to camp overnight for safety reasons - other vehicles may hit you. In areas where overnight parking is allowed you should park well off the road. Your RV unit must be completely self contained! Do not dump anything outside your RV, even water. 

--> Free campgrounds. There are free campgrounds located across the USA. These are very rustic, and will not have showers or flush toilets, but will have vault toilets. They may have a water spigot, or may not. They tend to be located in rural, out-of-the-way areas, not near tourist attractions. They may be on gravel or dirt roads. Tenting is allowed in free campgrounds. Here are two websites with many free campgrounds listed:

http://www.freecampgrounds.com/    and    http://freecampsites.net/usa/

... In general, when parking an RV overnight not in a campground, you should first consider the safety of your family and your RV. Get completely off any road and avoid areas with evidence of vagrants or gang activity. Be sure there are no signs prohibiting you from staying overnight. Be aware that some rural areas may have wild animals around, so use caution!