Eh, I think it's worth a shot to try walking--can you try walking for a few weeks and not lose the parking spot? I walk to work nearly every day--it's less than a mile for me, but it did take me some time to get used to it, and get proper gear for walking in winter.
Just start doing it tomorrow, right away, and see how it goes.
Do you absolutely have to be at work at a particular time each day? I walk but have a flexible schedule so if I show up at 7:15 vs. 7:30, it's no issue. I find that walking in the morning is a great way to wake up. You can also listen to music or audiobooks or podcasts, just like you would in the car, but I find it nice to be outside, enjoying the fresh air, sunrise, etc..
I'll admit I have driven to work (or gotten a ride) when it's below -10 F, because I've learned that's my temperature limit (and my husband is concerned I could slip on ice and freeze to death). I plan easy lunches I can bring to work--and if you ever forget lunch, it's a good excuse to order something in, or walk to a nearby place for lunch.
Also, co-workers know I walk to work and have offered to give me rides home if weather is bad--rainy, snowy, whatever. I have only ever taken someone up on this offer once in all my years of walking, but it's surprising how generous people are if they know you'll be hoofing it.