The whiskey on your breath Could make a small boy dizzy; (lines 1-2)
It seems like these lines are establishing that the drunken father has power over his son, who's being made dizzy, but think again. Isn't whiskey the most powerful player in these lines? It's influencing the father and making his son dizzy.
But I hung on like death (line 3)
This line shows us the most powerful thing mentioned in the entire poem: death. Even though the speaker's father seems so strong and masculine, death has power over him, and all of us, as Roethke found out at the age of fifteen when his own father died. Now, we can't be sure that the father and son in this poem are meant to be Roethke and his father, but we can guess that in writing about fathers, Roethke would be haunted by his father's passing.
My mother's countenance Could not unfrown itself. (lines 7-8)
The mother in this poem seems powerless as the kitchen gets wrecked. But she has the power of making her son feel guilty enough that he takes into account her displeasure even in the throes of waltzing with dad.