Milk makes us think of life and nourishment, so frozen milk (spilled milk?) is no good. It's like winter is trying to starve the people of this little rural community.
When blood is nipp'd, and ways be foul, (5)
If your blood gets too cold, or "nipp'd," you'll die. Death by freezing—heck, hypothermia—is always just beneath the surface in this poem.
Then nightly sings the staring owl, "Tu-whit, to-who!"— A merry note, (6-8)
Owls are cool and all, but they're hunters—birds of prey, that is—and thus potentially a symbol of death. It's perfectly fitting that there are owls in this poem because winter is the season of death (sometimes).
And coughing drowns the parson's saw, (11)
The "coughing" is fine and all, but right next to the word "drowns"? "Drowns" for sure makes us think of death, and "coughing" kind of does too, seeing as it's so close to that other word in the poem.