Father wears a dirty, oil-soaked monkey suit that cuts him under the arms, (7-9)
The apparent head of the family, and head of the station, wears a dirty suit that's too small for him. This might mean he doesn't care about what he looks like, or more likely, that he can't afford a better suit. This tells us the family might be a poor one.
and several quick and saucy and greasy sons assist him (8-9)
He's got sons! Several of them, and they're real slick kids and they're helping papa run the family business. We have a feeling that someday, they'll be sporting an ill-fitting monkey suit, just like papa.
(it's a family filling station), (10)
The speaker is perhaps stating the obvious at this point. Maybe that's why she's tacking on the parentheses here.
Do they live in the station? (12)
This is more speculative. We don't know, but we're about to find out.
a dirty dog, quite comfy. (18)
If a dog isn't part of the family, then we don't know who is!
Somebody embroidered the doily. (32)
Someone in the family has to put these homey touches on the place. Perhaps a mother figure? But if there is a mom in the picture: where is she? Has she left? Even died? Could that explain why things have gotten so shabby around the place?
Somebody waters the plant, (33)
Again, someone has to take care of the plants. So maybe mama's still on the scene, and she just doesn't get a center stage role in the poem. Either way, it's clear that this family isn't all doom and gloom and grease. Someone's taking care of the small stuff.