Any portable code that uses bitfields seems to distinguish between little- and big-endian platforms. See the declaration of struct iphdr in linux kernel for an example of su
The bit fields will be stored in a different order depending on the endian-ness of the machine, this may not matter in some cases but in other it may matter. Say for example that your ParsedInt struct represented flags in a packet sent over a network, a little endian machine and big endian machine read those flags in a different order from the transmitted byte which is obviously a problem.