You should be prepared for cold during a Yellowstone Park winter. (See my packing list for what to bring on your Yellowstone holidays in winter).
The northern part of the park, especially around Mammoth Hot Springs, is usually the warmest part of Yellowstone, though it can get well below zero F, there, too. While these average temperatures for Yellowstone in winter don’t seem that cold, trust me, it can get a lot colder. At least, that’s been my experience.
High/ low for Yellowstone in December: 26° / 4°F ( -3°/ -16°C)
High/ low for Yellowstone in January: 28° / 3°F (-2° / -16°C)
High/ low for Yellowstone in February: 31° / 4°F (0° / -15°C)
High/ low for Yellowstone in March: 39° / 10°F (4° / -12°C)