I think the decision is strictly your own. No right or wrong way, just do what makes the wife happy.
Of course, with another vehicle, you have all the upkeep with it: insurance, license plates, oil changes, maintenance and repairs. It's one more item to maintain, watch, fret over, deal with logistically (as you've indicated, even parking somewhere).
But, the advantage of having a run-around vehicle might out weigh all the headache responsibilities that go along with the 2nd vehicle. You have to weigh those headaches if they are worth it or not.
And then, nothing says you have to get rid of the 2nd vehicle RIGHT NOW! You can try, and see where it lands keeping both vehicles, having your wife drive the "chase car" for a while. It's possible, after a while, she may change her mind and realize the 2nd vehicle is just too much of a hassle. And then, on the other hand, you may come to terms with the idea that having the 2nd vehicle is actually very handy to have. So, my suggestion is to keep the 2nd vehicle for right now. Use it, see where the "comfort" level lays. And if it gets to be too much, then sell the second vehicle.
A lot of folks will get bicycles or motor scooters as a run-around mode of transportation too. Still your choice. Give it some time, try it, see how it works out. It's most important ... Keep the wife happy!