Both variations are correct, but
Ich reise im August nach Paris.
is more common or natural than
Ich reise nach Paris im August.
Another aspect that can influence the word order is emphasis. If you want to emphasise the point in time of your travel, you can use
Ich reise nach Paris im August (und nicht im September).
If you want to list your travel plans, it could also become
Im August reise ich nach Paris, und im September nach Rom.
As you see, while the word order subject - predicate - object(s) remains the foundation, it can be switched around if necessary.
As a side note, to use the verb "reisen" in this case feels a bit weird, more formal than necessary. In everyday speech, you probably would say
Ich fahre im August nach Paris.
meaning "I go to Paris in August" instead of "I travel to Paris in August".