1. This park is the best place to ride a bike.

2. This park is the best place in which we can ride a bike.

3. This park is the best place in order that we can ride a bike.
--------------------------------------
Does #1 mean #2 or #3?
Does 'to ride a bike' modify 'place'?

3 answers

"to ride" is the verb. It modifies nothing.
1 and 2 mean the same. 3 doesn't really make sense. We may know what is meant, but the words "in order that we can" are really meaningless.
I am too shorthand in the above. The predicate in the sentence, as you know, is "is". "To ride a bike" does modify place. "ride a bike" can be taken as the object of the preposition "to", a noun phrase describing an activity. "to ride" is also a verb. It does get confusing.
"To ride a bike" is an infinitive phrase.
http://www.chompchomp.com/terms/infinitivephrase.htm

It's serving as an adjective in sentence 1, describing "place."