Depending on what you actually want to say, you can connect the two independent clauses in any number of ways.
As you suggested works adequately, making the second a relative clause :
Two other families were also going on this trip, in which the parents agreed to look after Amy.
Making a time connection with while makes the second a dependent clause:
Two other families were also going on this trip, while the parents agreed to look after Amy.
You could choose any other subordinating conjunction from the list of common ones below to fit your meaning:
after,
although,
as, as if,
as long as,
as though, because,
before,
even if, even though,
if,
if only, in order that,
now that,
once, rather than,
since,
so that, than,
that,
though, till,
unless,
until, when,
whenever,
where, whereas,
wherever,
while,
A simple coordination with and:
Two other families were also going on this trip, and the parents agreed to look after Amy.
or but (or any other FANBOYS):
Two other families were also going on this trip, but the parents agreed to look after Amy.
You could use a semi-colon:
Two other families were also going on this trip; the parents agreed to look after Amy.
or a semi-colon with in fact:
Two other families were also going on this trip; in fact, the parents agreed to look after Amy.
And if that is not enough, you can also make the first clause dependent on, or relative to, the second.
Talk about flexibility!
