Here is a short list worth bookmarking. Most national parks keep dogs off the trails, so the ones that welcome a leashed dog are the exception, and they are worth planning a trip around.
Every park below allows leashed dogs on at least some of its real trails, not just the parking lot. We checked each one against the National Park Service's own pet page, and each links to its full rules.
A leash is required almost everywhere, usually 6 feet. Beyond that, the door is open.
ACADDog-friendly
CONGDog-friendly
CUVADog-friendly
GAARDog-friendly
JEFFDog-friendly
GRSADog-friendly
HOSPDog-friendly
INDUDog-friendly
KOVADog-friendly
LACLDog-friendly
MACADog-friendly
NERIDog-friendly
PEFODog-friendly
SHENDog-friendly
VIISDog-friendly
WHSADog-friendly
WRSTDog-friendlyIf a national park you want is not on this list, check its own page, then look at the national forests and BLM land nearby, which almost always welcome a leashed dog.
Shenandoah in Virginia and Acadia in Maine are the two standouts, with leashed dogs allowed on the large majority of their trails. Both are rare among national parks for how much ground a dog can cover.
The Park Service limits dogs to protect wildlife and fragile habitat, and because dogs can carry disease and disturb animals. State parks and national forests are usually far more open.