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If there is no contact after a week i would say there is no future relationship with said individual. Dont worry about specific amounts of time so much as quality of the interaction. It's ok to have one of experiences with people that don't ultimately lead to a friendship.
Did you just randomly meet this person at the park or was it prearranged in some manner?
Randomly met them.
Then they were probably just being polite and enjoying your company. It's not a bad thing, and I'm sure they enjoyed the time meeting you. But I wouldn't expect to ever hear back from them. Not every connection will be permanent, but that doesn't mean they didn't matter.