Hi, I've just restarted playing KSP after a long break and I'm having exactly this problem in Science mode. Using the - only available - MkI capsule; with a Commutron 16 antenna and a pair of symmetrically mounted Mystery Goo containers. (Oh, and I added the heat shield that the wiki tutorial doesn't mention.)
I've tweaked the chute to deploy at 0.5 atm/1,500m; I seem to be below 500m/s when the chute deploys and it's immediately destroyed. The chute indicator is white shortly after launch, but goes to red once I'm too high to use it and remains red until I'm well into the atmosphere again after all the heat and noise of re-entry - until shortly before it deploys in fact. I have the chute staged as the only item on Stage zero and I leave it to deploy automatically as above. So I am about 99% sure that the chute is not deploying early. Is this a glitch in Science mode? I'd really like to know how to fix it.
Thanks to those who offered help. I actually took some advice from nobodyhasthis and installed RealChute (so thanks to StupidChris for creating that). Now I have pre-sets that act as I would expect chutes to act, but I also have more information available so that I can design custom chute set ups for myself later on. I can truthfully say I couldn't manage without it.