no - once Voldemort inadvertently destroyed the Horcrux that Harry carried Harry lost his connection with Voldemort.
Actually its arguable, even though Harry had the gift through his connection with Voldemort, Ron was able to speak parclemouth in the final book without doing so previously just by repeating words so technicaly if harry remembers the words he could. Though he will never be able to do it as well as he did before.
It is speak parseltongue. The person who speaks it is a parselmouth.