Whatever option you choose, don't forget to compare its cost against the cost of airport parking, which usually isn't cheap. It's possible (but not guaranteed) that you could save money with a taxi rather than parking for a week.
Check to see if there is a car rental company with an office near you and the airport you'll have to fly out of. I've done this, and it's pretty convenient. You can rent a car for the trip to the airport from a location near your home, return it to the rental company at the airport you'll depart from, and then rent another car at your destination for the duration of the trip, then do the same in reverse when you come back. Only one car is rented at a time, and only at times that you're actually using it.
If you don't have that option because there isn't a car rental company with appropriately located offices, consider public transit or long-haul buses (like Greyhound). Both will probably be less expensive than a personal taxi for a sizeable trip, though scheduling might be problematic. You could also try a shuttle service (many airports have affiliations with these), which are usually less expensive than a taxi and more flexible than a Greyhound or public transit bus.
Finally, if your only workable option is to drive your own car to the airport and leave it, but you don't want to do that, recognize that your employer doesn't have any particular reason to accommodate your personal desires for your personal property in this case. If the only thing stopping you from using your own vehicle is your own preference to not do so, it's not unreasonable for you to bear the cost of satisfying your own preferences (which have no bearing on your employer).
The least expensive way to do that last one would be to have a friend drive you, in their car or your own, so that no one has to park. A friend will usually do that for free (I've done it for free plenty of times!), but if the trip is really onerous a token payment can sweeten the deal-- I've gotten favors from friends in exchange for drinks, or promises of reciprocal favors down the line.