The first sentence
National parks are parks that are run and managed by the Federal government.
is your safest bet, at least in the United States, because National Parks (notice the capital letters) are not necessarily the only "parks" run and managed by the (United States) Federal government.
Use a definite article before a noun or noun phrase that already uniquely identifies what it is referring to. With a common noun, you usually introduce the thing with an indefinite article, then use a definite article later:
I visited a park in Millbury today. I don't remember the park's name.
But you can use "the" for a specific, well-known thing
I went to the park today (that is, the park we always go to)
or a proper noun that might be confused with a common noun
The Federal Government (if I'm in Australia, I'm talking about what goes on in Canberra)
The key difference here is that the designation "National Park" means something specific about an area in the nomenclature of the National Park system (which you can read about here if you want to spend the time).
But "park" is a common noun, and doesn't even have to mean a "park" facility maintained run by a government. For example, South Park and its siblings in Colorado are just geographic regions.
In some other countries where the system is set up differently, you might correctly use the second sentence, (adding another definite article at the beginning):
The National Parks are the parks that are run and managed by the Federal government.
or, select parks with "those":
National Parks are those parks that are run and managed by the federal government.
However, the first sentence, without any extra articles, is always correct because it claims less.