I'm assuming the adverbs have to go in that respective order in those clauses.
Adverbs are complicated in English. They perform a variety of functions. I won't go into details about the various functions, but their distribution (placement) can vary depending on the function of the adverb.
I'd say the intended answer is:
He obviously can't go with you on that trip, because his parents are definitely against it.
Obviously, he can't go with you on that trip, because his parents are definitely against it.
Obviously here is an evaluative adverb; it provides the speaker's evaluation of the proposition (his inability to go with you). Evaluatives can go in a number of spots (two are shown above), but not within any phrases.
The position you had it in is between the auxiliary verb can't and the lexical verb go, which is squarely in the middle of the IP (or VP, depending on your theory). In this position, obviously is still functioning as an evaluative, but the proposition that it evaluates is his going, not his inability to go, and this whole evaluated chunk is then negated with can't. To bracket this out:
He [can't [obviously [go]]]
That is obviously modifies go whereas in my answer above:
he [obviously [can't [go]]]
Go is negated first, and then the negated chunk is evaluated.
So the meaning of he can't obviously go is parallel in my opinion to you can't seriously mean that. But the meaning of obviously doesn't exactly lend itself here.
To show the difference, a better adverb would be clearly. Consider the two sentences below:
John doesn't clearly understand the task.
John clearly doesn't understand the task.
The meaning of the first is that john doesn't understand the task very clearly. So he understands it to some degree, but not enough. Clearly modifies understand.
In the second, John doesn't understand the task at all, and clearly provides the speakers evaluation of the proposition: the fact that John doesn't understand the task is clear.