parted documentation, the
mkpart command creates a partition without creating a filesystem on it.
You might or might not need to run
partprobe afterwards, depending on the versions of the kernel and
parted used. Older versions might need it, newer ones generally won't. However, running it should not be harmful in any case.
But if you want to keep the partition you just created, your
mkfs command should then be:
mkfs.ext4 /dev/sdc1 # not /dev/sdc
If you wanted to use the disk in a so-called "superfloppy" configuration, it is certainly possible to just run
mkfs on the whole-disk device
/dev/sdc and use it like that. But then there would be no point in partitioning it first, as creating the filesystem like that will overwrite the freshly created partition table.
Having a partition table on the disk that is recognizable on most common operating systems makes it safer to move disks between systems: it avoids the possibility that another operating system (I'm looking towards Redmond...) would not recognize the disk as already containing data, and might offer to helpfully format it.