Je ne peux pas vendre que des parties de la voiture.

You are dealing with a “ne … pas … que …” construction here. Without “pas”, “I can only/just sell parts of the car, but not the whole thing”. By adding “pas”, you can negate this “only/just” restrictive sense: “It is not possible for me to sell just/only parts of the car”.

1) You cannot move “ne” and “pas” from their respective current positions.

2) The word “que {only/just}” in a restrictive sense always comes immediately before the word(s) that you want to qualify; the phrase “des parties de la voiture” is the very idea that you want to qualify with the restrictive “que”:

not … just/only parts of the car

(=) ne … pas … que des parties de la voiture

Je ne peux pas que vendre des parties de la voiture.

If you place “que” right in front of the verb “vendre”, on the other hand, you are now imparting a restrictive sense to the act of “selling”:

I cannot just sell X, (but also have to do other things with it: offer an after-sales service etc.)

The idea enclosed in parentheses is implied.

1. Je ne peux pas vendre que… is just the negation of Je ne peux vendre que…
   Je ne peux pas ne que vendre is agrammatical.
   Here are some examples:

• Il n’a pas bu que de l’eau.
• Je repensais à tous les mirages que j’avais ainsi découverts dans notre appartement et qui n’étaient pas qu’optiques. (Proust)
• Sa boutique n’était pas qu’un lieu de rendez-vous. (Céline)

2. Here, que is still part of the ne … que splitted negative so doesn’t stand on its own but there are a few standalone que cases, e.g. :

• Aucun lieu pour se distraire que le cabaret. (No entertaining location but the cabaret)
• Une leçon qu’à moitié sue. (A partially assimilated lecture)
• Héloïse aime les pygmées albinos, mais pas que. (…but that’s not the only people/thing she likes)

Examples and references from gabrielwyler.com:
– Ne…pas que… et ne…pas…que…
– Que absolu