The phrase "allow them to manipulate" is correct and usable in written English.
It can be used in contexts where you are discussing giving permission or enabling someone to exert control or influence over something or someone.
Example: "The new policy will allow them to manipulate the data to better suit their needs."
Alternatives: "permit them to control" or "enable them to influence".
"You allow them to manipulate you.
The Mets have the next three Mondays off, which could allow them to manipulate their rotation so that Lawrence does not pitch every turn.
Where the consultants of the nineteen-sixties and seventies obsessed over how to use television to beam ideal images of their clients into voters' homes, today's spinmasters hope that big data will allow them to manipulate voters' deepest hopes and fears.
The challenge for creators of earthbound robots has been to develop the bots' artificial intelligence and mechanical dexterity to allow them to manipulate objects and respond to a complex environment.
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The reason we allow people to walk all over us, or allow them to manipulate or coerce us into doing something is complicated.
That allows them to manipulate prices, and to retain their grip on the fractious OPEC cartel.
They fear that the traders' ownership of infrastructure allows them to manipulate local prices, even if they do not have the heft to rig global markets (an American regulator accused the unit JPMorgan Chase sold to Mercuria of such behaviour last year, and extracted a $410m penalty).
The system allows them to manipulate images with their voice and hand-gestures rather than using a keyboard and mouse.
The psychologists also created a replica of the room in virtual reality, allowing them to manipulate the position of furniture, doors, and windows.
I also wanted to give the DJs the best platform to perform on, giving them access to cutting-edge kit which allowed them to manipulate the system and, with a good lighting engineer, to accentuate the music visually.
