The Polish world No. 5 defeated the 66th-ranked Parry 6-3, 6-3 in the women's singles round of 16, producing some of her best tennis of the tournament.
Świątek, who is the defending champion in Cincinnati, will next face the winner between Kazakhstan's Elena Rybakina and Russian Diana Shnaider for a place in the semifinals.
The Cincinnati Open hard-court tournament is the final warm-up for many top players ahead of the US Open Grand Slam, which starts on August 30.
Earlier this month, Świątek won the Canadian Open in Toronto, claiming her 26th career title.
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Source: IAR/PAP