Agreed, but how does that alone make Dumbledore decide it's a good idea?

Plus, Dumbledore doesn't want to take Harry to the hall to hear his own prophecy, and this is most likely because he knows there are other prophecies that Harry would hear. And he never actually confirms that he got his information from James and Lily, Harry comes up with this and Dumbledore let's him believe it. I think Dumbledore had other access.

"The Hall of Prophecy," Albus confirmed lowly. "Those who are spoken of in a prophecy, may listen to that prophecy there. Do you see the implication, Harry?"

Harry frowned. "Well, I could listen to it, or the Dark Lord... oh, my parents. Those who had thrice defied him. They were also mentioned in the prophecy, so they could hear the recording?"

"If James and Lily heard anything different from what Minerva reported," Albus said evenly, "they did not say so to me."

"You took James and Lily there? " Minerva said.

"Fawkes can go to many places," Albus said. "Do not mention the fact."

Harry was staring directly at Albus. "Can I go to this Department of Mysteries place and hear the recorded prophecy? The original tone of voice might be helpful, from what I've heard."

Light glinted from the reflection of Albus's half-moon glasses as the old wizard slowly shook his head. "I think that would be unwise," Albus said. "For reasons beyond the obvious. It is dangerous, that place which Merlin made; more dangerous to some people than others."