It was the Toshiba drive that was causing the problem. It's not that it spins down, but it unloads the head, and it has a delay when it loads it again.
According to Toshiba technical support, the DT-series of harddrives do go into standby mode when inactive. They also state that "this cannot be circumvented". I got a recommendation to instead look at their enterprise drives, which hasn't got this delay. However, it turns out that using hdparm's -B and -S options as Hennes suggested in the comment below does prevent head parking, though these commands of course have to be sent to the drive after each power cycle. Since the computer in question is a Windows 8 system used for music production, the solution was to use a hdparm distribution for Windows and run it as a service at each login.