Hickman and Parry finished third and fourth respectively in their heat to comfortably qualify for the semi-finals which take place later on Monday.
Parry went off quickly and led after the first 50m, but American Tom Malchow stormed through to win the heat in a new Olympic record time of one minute 56.25 seconds.
Hickman was edged out by Australian Justin Norris for second but came home just ahead of Parry, who tired somewhat on his final leg.
"It was just a matter of getting the nerves out of my system," Parry said. "I think I maybe went out a little too hard and I faded over the last 20 metres at the end.
Hickman added: "I'm quite pleased. I'm not that far off my best."
Home favourite
Elsewhere in the pool, the morning session saw home favourite Susan O'Neill set the pace in the women's 200m freestyle, winning her heat in 1:59.14.
The only other swimmer under two minutes was South Africa's Helene Muller, who clocked 1:59.89, in a heat in which karen Pickering could only finish seventh.
Her time of 2:01.42 was 0.11 seconds outside the qualifying mark for the semi-finals, although she refused to be too disappointed.
"I'm quite happy with the time even if I'm a bit disappointed with my placing," Pickering said afterwards.
"There's still the 100m and the 4x200m relay so I've still got a lot to come."
Fellow Briton Karen Legg actually missed the race because of a flu bug.
Meanwhile, Britain's Badminton duo of Simon Archer and Nathan Robertson lost at the quarter-final stage of the men's doubles to Indonesian duo and top seeds Tony Gunawan and Candra Wijaya 15-13 15-11.
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