I often visit the park for family picnics.
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What does 'for family picnics' modify? Does it modify 'visit'?
If the park is allowed only for family picnics, can it modify 'park'? This park is a special park. It is used only for family picnics.
If the prepositional phrase modified "park," it would tell which, what kind of, or how many.
The phrase modifies the verb "visit." It tells why they visit the park.