Sounds like sulking to me. From Oxford Dictionaries:
VERB [NO OBJECT].
Be silent, morose, and bad-tempered out of annoyance or disappointment.
Some other definitions emphasize that the silence might be attention-seeking.
So you could say
Dad sits back in his seat, and sulks. "Fine," he says.
(Coincidentally, the example sentence for sulk in Google's boxed definition at the moment is "Dad was sulking," apparently a shortened form of one of Oxford's examples.)
Note that while I wouldn't hesitate to use it in your situation, it's quite likely that "Dad" would find the description insulting, as there are some connotations of childishness. I'm pretty sure I've read somewhere (probably dialogue in a novel) something like "children sulk; men brood"1 or "I [as an adult/a manly man] don't sulk, I brood." This term might suit if you want to be more diplomatic:
- VERB [NO OBJECT] Think deeply about something that makes one unhappy, angry, or worried.
This works especially well if you want to emphasize Dad's inner thoughts (something like "Dad is ruminating on his anger"), whereas sulk is better if you want to emphasize the outer behavior ("Dad is sitting in quiet anger").
There are many synonyms for sulk that you might want to look into, though I think sulk is likely the best for of those in common use.
Since you mention in a comment
I could have sworn it started with a "G" like glowered, but I don't know. I only ran into that word once. I thought it was some obscure word you'd need to memorize a 2,000,000 word dictionary to know it even existed.
I'll point out that there really are a remarkable number of gl-/gr- terms with similar or related meanings, such as gloom, glum, grump, grumble, grouch, grouse, groan, grizzle, and your own glower (all of which can be used as verbs). Browsing through the Oxford English Dictionary turns up some more obscure examples, like glump, glout, grumme, and grutch. Perhaps one of these, or another, similarly obscure term is what you recall.
If you're open to a slight rephrasing, huff might work in one of its forms:
Dad sits back in his seat in a huff. "Fine," he says.
Dad sits back in his seat. "Fine," he says with a huff.
Dad sits back in his seat. "Fine," he huffs.
And so forth.
1 Which, all those brooding anti-heroes notwithstanding, is kind of ironic, etymologically.