Word to describe to sit in quiet anger
Solution 1:
As others have mentioned in the comments, the word that my dialect uses for restrained-but-hot-with-anger is
Stew
"We need to travel all the way to Montreal to pick up my sister," Mom says.
Dad sits back in his seat and stews.
"Fine," he says.
It's worth noting that this might be an Americanism. MW provides the appropriate "be in a state of suppressed agitation, worry, or resentment", but Oxford Dictionaries considers stewing to only be a synonym for worrying.
Stew's a step under boiling, which is a step under Mr Grimm's fuming, which is a step under ab2's seething, which is properly a much more violent emotion than is appropriate with your example, unless Aunt Agnes crashed Dad's Honda while giving roadhead to her flavor-of-the-month boytoy the last time she was in town and then told the teenagers about it. Sulking is for little kids and suggests Dad is being petulant and petty in his annoyance.
For what it's worth, there are many more descriptive actions that would serve you better if this were some kind of short story. Glaring, staring out the window, rolling his eyes, sighing, counting to ten, 'deadpans' instead of 'says'... any of them work a little better than a word for having restrained anger. Those words will be more appropriate for describing his mood later during the actual trip.
Solution 2:
seethe, from Collins English Dictionary
When you are seething, you are very angry about something but do not express your feelings about it.
She took it calmly at first but under the surface was seething.
She put a hand on her hip, grinning derisively, while I seethed with rage
In the OP's example:
Dad sits back in his seat, seething or
Dad sits back in his seat and seethes.
Solution 3:
Fume is another possibility. It can be used to describe someone being angry in a way that refers to their internal mental state, rather than their actions or behaviour, as described in the question about sitting in quiet anger.
From Wiktionary:
To feel or express great anger.
He's still fuming about the argument they had yesterday.
The literal meaning of "fume" is to emit fumes, and can mean something having an internal fire that you can't see and you only notice the fumes coming out.
Solution 4:
One such verb is simmer.
Dad sits back in his seat, and simmers. "Fine," he says.
ODO:
simmer
VERB1 (of water or food) stay just below the boiling point while being heated.
‘the goulash was simmering slowly on the stove’
[figurative] ‘the disagreement simmered for years and eventually boiled over’1.2 Be in a state of suppressed anger or excitement.
‘I simmered a bit and put her remark down to her lack of understanding of classic cars.’