I am not a teacher and my English is not good. Here are my two cents for your reference only.
1. My parents shipped/packed me off to the military school
I think it is okay. They are kind of metaphor. It would be exactly right if you had to go to the school by sea.
2. Could I use " Sent me to the military school" instead of the examples above?
Yes, you can. I think it is suitable for any situation and more common.
3. At the age of fourteen, My parents shipped/packed me off to the military school.
I think there's a mismatch between " At the age of fourteen" and the main sentence. The subject is " my parents" and it is impossible for them at 14 at that time. I try to touch them up as belows.
1) When I was fourteen, my parents sent me to the military school.
2) At the age of fourteen, I was sent to the military school by my parents.
Hope teachers here or other forumites can give you a hand genuinely.