Why can't you say 「一人と」「あるいてで」公園へ行きます？
If you have learned Japanese long enough, you might have wondered why you can't say 「一人と」and 「あるいてで」行きます, though you can say that 「友達と」or「バスで」行きます。
The reason is that you need to form て- form of the noun to connect with verb (行きます)
To form て- form of Noun, just add 「で」. So て- form of 「一人」is「一人で」.
And speaking of 「あるいて」, the reason is the same. That means you need to form 「て- form 」again to connect verb ( 行きます）. And て- form of 「あるきます 」( walk ) is「あるいて」.
Difficult? I think so. But after you become intermediate Japanese speaker, I bet you will understand that =)