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Related to parapets: parapet wall

keep (one's) head below the parapet

To hide or remain inconspicuous in an attempt to avoid trouble or danger. I wouldn't say anything right now—just keep your head below the parapet for the time being.
See also: below, head, keep, parapet

put/stick your head above the parapet

to be brave enough to state an opinion that might upset people Kearton was one of the very few to put his head above the corporate parapet and speak his mind in public.
See also: above, head, parapet, put