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.
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Does #1 mean #2 or #3?
Does 'to ride a bike' modify 'place'?
"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."
Both sentences #2 and #3 provide clarification on where we can ride a bike, but they have different sentence structures.
In sentence #2, "in which we can ride a bike" is a relative clause that modifies "place." It clarifies that the park being referred to is the best place for bike riding.
In sentence #3, "in order that we can ride a bike" is an adverbial clause that explains the purpose or intention of the park being the best place. It suggests that the park is especially notable because it allows bike riding.
Regarding whether "to ride a bike" modifies "place" in sentence #1, it does not directly modify "place." Instead, it specifies the purpose or activity that is best suited for the park. It implies that the park offers the most favorable conditions or amenities for bike riding.