MSR uses the construction 'mod i'n a lot . It seems to mean ‘… that I’m…’ and therefore possibly a form of ‘bod’, but I don’t see where the contraction comes from, and I can’t find it in a dictionary.
Also, why is it barcio'i gar rather than barcio'r gar?
FWIW, my best guess at a rough translation is:
He knows, at some level, that I’m laying down the law to him over parking the stupid big car outside someone else’s house.
The SSiW course teaches a “slangier” (but completely acceptable in informal speech) version – bo’fi. The grammatically correct construction is fy mod i’n, and MSR simply drops the possessive particle fy, with the conventional apostrophe to denote that something has been shortened → 'mod i’n
This structure changes depending on the person in the subclause, and the mutation patterns reflect what you expect in possessive structures: Dw i’n meddwl dy fod ti’n gywir. - I think that you are right. (and ei fod e’n / ei bod hi’n respectively).
It was the ' which threw me – I’ve only ever noticed it used for a contraction of the word it begins (or ends), not to signal that another word has been missed completely, so I was looking for a longer form of bod and couldn’t think of one.
Regarding the car, your translation is nearly right but it’s, “I’m laying down the law to him over parking his stupid big car outside someone else’s house.”
“Barcio’i gar” is a contraction of “barcio ei gar”. The possessive “his” causes the soft mutation from “car” to “gar”.