It is believed that a letter arrived at Anfield on Thursday - just 48 hours after Liverpool's triumphant passage into the Champions League semi-finals - demanding chief executive Parry's departure.
Parry is currently in London at a Premier League meeting but he has been made aware of the communication and is expected to refuse to quit by return of post.
Gillett's son Foster was also in attendance on Tuesday, as was Hicks' son Tom Jnr, both sons also being directors of the club.
And it is also believed that Liverpool fans groups have received e-mails telling them that Hicks wants Parry out.
However, with the club ownership split 50-50 between the two Americans, Hicks would not have been able to have got sanction for Parry's dismissal, hence the letter received at Anfield today.
Hicks was believed to have been enraged by Parry's interview recently on BBC Five Live in which the chief executive pleaded with both co-owners to end the war between them and come to an agreement that would allow the club to move forward.
It was Parry who was largely involved in bringing Gillett to the club 18 months ago, with the American bringing in Hicks for financial muscle to get the deal through.