Both fired six-under-par 66s to finish day one in front of Wales' Sahra Hassan, American Cindy Lacrosse and Czech player Klara Spilkova.
The event is the last of the Ladies European Tour season and Order of Merit leader Carlota Ciganda began with a 70, while German Caroline Masson, £26,000 behind in second place, had a 68 to be in a share of sixth spot.
Masson needs a top-three finish to have a chance of stopping 22-year-old Spaniard Ciganda from becoming the first player since Laura Davies in 1985 to win both the money list and the Rookie of the Year title in the same season.
Parker, who has been practising in Dubai for the last two months, birdied the par-five last to catch Feng in the tournament, where she has finished ninth and fourth the past two years.
"I do usually do well on this course - it suits me," said the 23-year-old, whose brother Ben, a Challenge Tour player, is caddying for her.
"My putting was really good. I hit 14 greens and had 26 putts. I feel so comfortable here."
In June, Feng became the first Chinese golfer - male or female - to win a major when she captured the LPGA Championship, but this is her debut in Dubai.
"I came with pressure actually," the world number six said. "Before the round I told myself I have to be patient, whether I'm playing well or not.
"I was only one under through seven holes and missed a lot of chances, so to finish six under I'm really happy about that."
Masson came home in a four-under 33 to re-ignite her money list title hopes, grabbing birdies at the 10th, 12th, 13th and 515-yard 18th.
Seventeen-year-old defending champion Lexi Thompson had a 69, while fellow American Michelle Wie shot 70 and Davies 71.