The 23-year-old Paratore, chasing his second title on the European Tour, sank nine birdies with gains in the final four holes before Norris too finished the round strongly.
"Today the short game was really good and also the iron play," Paratore said. "The last four holes I holed some good putts on 18 and 17 so I'm pretty happy with the short game."
Norris, who lost his father in July, said he was enjoying his golf again with his brother carrying his bags.
"It's been a rough year last year after losing my father... it feels like everything is just working at the moment," Norris said. "We're having a lot of fun out there, that's the main thing, I'm happy with what's happening."
Four-times major champion Koepka, who has not played since slipping on concrete in the PGA Tour's C.J. Cup in South Korea in October and injuring his left knee, shot a bogey-free 66 to sit alongside Australian Jason Scrivener.
"I like the way I've played. I've kind of known I've been hitting it really well, putting it really well for a couple of weeks," Koepka told Sky Sports. "It's good to be back. I've missed the competition.
"I think the first day I picked up the club... I felt like I hadn't left. I've done it for years and years. You don't forget how to swing the golf club.
"It (knee) feels fine. It was a little sore last night, just did some treatment on it, that's expected. This is a first week I've walked 18 holes and I've done it three times already. It's a little tired."
Spaniards Rafa Cabrera Bello and Sergio Garcia were in a group of players tied for fifth while holder Shane Lowry was further back after an opening round of 70. (Reporting by Shrivathsa Sridhar in Bengaluru Editing by Christian Radnedge)