Mm, no, the opposite I think – the rider will fall slowly. The spring remains extended due to the weight of the rider – being held aloft by a tower vs. a parachute makes no difference here. However the descending parachute shortens the spring at a rate slow enough that it can be transmitted to the rider well before the parachute reaches the ground, so the rider will fall a little bit.
(This is analogous to simply lowering the hand holding the slinky – the bottom eventually follows suit.)
Consider the slinky again. If I dropped it with a parachute to slow it.... no, replace "parachute" with "my hand", but assume my hand follows the same trajectory the parachute would... the end of the slinky would not wait for my hand to approach it. The information about a change (the drop) would take the same amount of time to reach the end of the slinky. The information travels just as fast, even though the information is "we're falling slowly" instead of "we're falling".
(This is analogous to simply lowering the hand holding the slinky – the bottom eventually follows suit.)