Yes, any chassis part upgrade is carried over to your Career mode F1 car.
Let's say that you already own the Mercedes F1 car for the Duel/Event mode and its chassis part is level 4, then, when you will sign your Master contract with Mercedes in Career mode, the F1 car lent to you by the team will already be level 4 even though you haven't completed any GP yet with their F1 car.
The opposite is also true. If you don't own the Red Bull F1 car and if in the Career mode you have upgraded your Red Bull chassis part to level 3, then, if you buy the Red Bull car in the Shop, it will already be level 3 as it carries over the progress of the car made in Career mode.
As a reminder, to be able to change your equipped chassis part for Duel/Event mode, you must first own an Official car. These can be purchased using Coins or real money from the in-game Shop.
Also, any performance gained by installing parts on your car isn't reflected in the performance of your car in Career mode. In Career mode, your car's performance is calculated solely on the basis of the level of your car.