I'm going to apologize in advance if it seems like I don't believe you all are doing everything, but I've noticed something that I don't understand as it relates to "Home" in contacts that may be creating a problem, and it would be real easy to miss. So forgive me for going step by step.
There can be a disassociation between a contact card called "Home" and the use of "Home" as recognized as a location of significance on the Iphone. Don't know if that relates to the fact that I use Exchange for my contacts, but I don't think so.
You can see if that disassociation exists by the following. You should see a listing at the very top (above the A's) that will say "my card", and if you click on it your address info should be there. Then, scroll down to the actual listing. Next to the contact used to create your Home Location, it should say all the way to the right "me". If either of those things are not present, there may be a dissociation between your Home Location as set in the IPhone and your contact card for Home.
Whether you see the disassociation or not, I recommend you try the following.
If you do not have an entry in your contact list called "Home", create one. Be sure to include a physical home address. Then go to "Settings", "Contacts", and hit the "My info" listing. That will take you to your contact list, scroll down to the "Home" contact card, and select that.
Now go to your home screen, double tap the button to pull up a list of all "open" apps, and close all of them, particularly the "contacts" app.
Now reopen your contacts. You should have a "my card" listing above the "A's". Scroll down to the Home listing in the "H's", and be sure the contact card listing for "Home" says "me" on the right side. At that point, your IPhone location and your contact card are synced. Now see if the car location works.
I'm not sure if you really need to have a separate card for "Home", or if you can just point the "My info" entry above to a named contact card, but I'll leave it to others to test that. For now, however, use a card called "Home" to see if that will work for you.
Two things to note. Fully closing the contact app (double click the home button and "flick away" contacts) is important, as the contact list will not fully update until you close and reopen the app. Just going to the home screen and back to contacts is not sufficient. Second, when testing to see if this works, note that your parking location does not pop up immediately. I don't know if it is a time delay, or if after noting your location when you turn off the car and terminate the bluetooth connection, it does not display until you have physically moved away from the vehicle. Either way, when testing, after turning off the car walk 30 feet away and then see if your parked location is noted.
Your issues may be different, but I have been able to "break" and then "fix" this function on my Iphone by screwing around with these entries, so it may be worth a shot for you to try.
Good luck!