Optimal sensor for measuring object velocity in an open area?
GPS is easiest to implement with lots of off-the-shelf hardware available. With a rugged (shock resistant) receiver, it will be reliable. GPS raw data is cartesian coordinates in earth centered inertial reference frame, receivers convert this to lat/long/height in earth fixed frame (see WGS-84). Postprocessing will allow you to convert to any reference frame you may want.  

GPS receivers evaluate distance changes to space vehicles at a high rate, often at 1kHz. Even better, these deltas are more accurate than the position solution (because of ambiguities). Unfortunately, the raw pseudorange data and rates is often not available from consumer grade equipment. The position solution in contrast if often done only at 1Hz, better units do 10 solutions per second. To make things worse, solvers do low-pass filtering on acceleration. This makes the output look better (less noisy), but it is less accurate. If you want accuracy, do not calculate deltas from solutions, instead go for raw pseudoranges and doppler rates (RINEX-format is the keyword to look for).

Weather conditions will probably not be an issue, I assume your parachute is in open sky? Parachute fabric should be transparent to microwaves (is this also true for wet fabric?). The patch antenna should be placed on top to maintain satellite visibility even when spinning.

Jerks on high speed chute deployment can be an issue, your receiver may temporarily loose lock on satellites if line of sight dynamics exceeds \$10m/s^2 \$ (this is due to limited loop bandwidth in the tracking loops and applies even when using raw data).