“at” and “to”
I went on a trip at the zoo.
means that while you were at the zoo, you went on a trip. For example, maybe you took a ride on a little train that travels through the zoo.
I went on a trip to the zoo.
means that the zoo was the destination of your trip. For example, maybe you and other people in your school took a ride on a bus and visited the Columbus Zoo.
One of the primary meanings of at is to indicate a location considered indivisible, and one of the primary meanings of to is to indicate the destination of motion. But don't think that these prepositions mean the same thing in every context. Often with English prepositions, you need to understand the phrase as carrying an important part of the meaning, which can't be inferred from the words and grammatical rules alone.
In this sentence:
I went for a coffee at my parents'.
a listener tends to hear coffee at my parents' as a separate, familiar phrase. In this context, coffee means spending some time together drinking coffee and talking. The word at still carries its meaning of indicating a location. The phrase a coffee, which overlaps coffee at my parents, suggests that you drank only one cup of coffee.
I went for a coffee to my parents'.
sounds a little strange, but it's still grammatical. The reason for the strangeness is not a grammatical rule, but the familiarity of the phrase coffee at location. The word coffee is an important part of the familiar phrase, though many kinds of social activities can fit there.
Competition from other familiar phrases
By the way, without the apostrophe, this sentence:
I went for coffee to my parents.
means that you asked your parents for coffee, probably because you had run out of coffee and you couldn't buy coffee for yourself. That is because this sentence doesn't contain the activity at location formula, and it makes a listener recall two other familiar phrase patterns, which interfere with the meaning you intend. One of them is:
I went for help.
This means that you searched for someone to help you, probably traveling away from your current location. The other relevant phrase pattern combines this one with for:
I went to a bank for a loan.
I went for a loan to my parents.
As you can see, in this phrase pattern, the prepositional phrases can be reversed. The went for phrase often suggests seeking help from someone else. In that context, to indicates who you are seeking help from or where you're seeking it.
There is actually no great danger of being misunderstood if you vary a preposition in a familiar phrase. People use context and common sense to infer your meaning. But the above shows the role that that prepositions commonly play in English, which goes beyond their meanings as individual words: they lead a listener to recall one familiar phrase pattern or another, each of which has its own meaning.