I think you are
At 1 I was born when my parents parted,
And no folds were used. So that's how I started.
P - which is the first letter of parents and parted is also the voiceless bilabial plosive, which means that the sound does not use the vocal cords, a.k.a. vocal folds. Each lip can be thought of as the parents of the "P" sound since they are responsible for the plosive (stopping of air flow) part.
At 2 I was famous since ages ago,
But looked more like three and continued to grow.
Pi - a famous number since at least Ancient Greece which is a little bit greater than 3. In ancient times, Babylonians, Hindus, and Chinese used the value 3, but as time went on, the value grew in accuracy and is still growing today.
At 3 I was cover for some compressed files
To get to your email and cause you to smile.
Piz - a file extension representing a compressed .zip archive. This is often used for hiding the file type when sending zipped email attachments so that servers don't block the message.
At 4 I was nothing, so don't even start,
But add "LE"- I'm an Aussie private part.
Pizz - which means nothing by itself but pizzle is indeed slang for a private part.
At 5 I was food- so diverse and delish!
Shape me, refashion me- however you wish.
Pizza - indeed, a diverse, delicious a customisable food.
At 6 I really became nothing again
But I'm thankful so far- just look where I've been!
Pizzaz - which means nothing by itself, although some dictionaries list it as an alternative spelling to pizzazz.
At 7 I was exciting, in fashion.
I'm glad that I finally found my passion.
Pizzazz - the quality of being exciting, attractive, fashionable or glamorous.