Is Yosemite Valley Part of Yellowstone National Park?

No. They are separate parks in different states, ~750 miles (1,200 km) apart:

• Yosemite Valley → Yosemite National Park → California (Sierra Nevada)
• Yellowstone National Park → Wyoming (96%), Montana (3%), Idaho (1%) — mostly on the Yellowstone Plateau

Both are UNESCO World Heritage Sites and crown jewels of the U.S. National Park System, but they are distinct in geology, ecology, and geography.