parallelism in a sentence

You should have a list of verbs or a list of nouns, so your second statement, with the list of verbs, is good. You could have a list of nouns with "A public authority is looking to improve productivity, transparency of information and management of public funds". The alternative in the comment is another possibility, but of the three I prefer your second statement.

Note that "looking" in this context means "intending" or "aiming", so it is correct without being changed to "looking forward".