What is your processor? PLS71 doesn't match any AB processor that I'm aware of - the closest thing I can imagine is a 1756-L71, but I don't believe it could be one of those as they only support version 20 firmware at the earliest, from my recollection. I don't know of any processor that supports v16, that also supports anything higher than v20, so I'm not sure what processor you have that could be flashed from v16 to v32.
All that said, it probably doesn't matter. The more important question is, why do you want to flash from series A to series C on the 1734-AENTR? Unless you specifically need some features of series C that are not available on series A - and to my knowledge there generally aren't any - you can define it as whatever series you like, make sure that "compatible module" is selected under electronic keying (which it is by default) and the system will work with just about any series.
To answer the question you actually asked - no, I don't believe you can update a module from Series A to Series C. The series, as far as I'm aware, represents a hardware version - a series C is physically built differently at the hardware level. You can flash the
firmware using controlFlash - though there are limitations on which firmware versions are available, based on which hardware version (Series A/B/C) you have.
This technote (techconnect required) goes through the different firmware compatibilities for the different series, and has instructions on how to locate and flash the firmware.