Physio Parry, who has been at Prenton Park for over 20 years, combined with Allison and reserve and youth-team coach Shaun Garnett to guide Rovers to only their third win of the season at the weekend in what was the regime's second match in charge.
Parry insists he is enjoying the experience but told the Liverpool Daily Post: "Whether I am enjoying it enough to want it long-term is another matter. At this moment in time, probably not.
"It is difficult because I am doing two jobs and that does not help. All of my days are packed at the moment. There is all the peripheral stuff the manager has to do - talking to players who aren't happy and things like that. It doesn't bother me but I don't particularly enjoy it."
The Rovers hierarchy are currently considering their options having sacked John Barnes earlier this month and Parry wants to see colleague Allison, widely known as 'The Chief', go for the position.
He added: "The Chief was talking to me about applying for the job and was a bit embarrassed. But I said 'put in for the job'. It makes no odds whether I want it or don't want it, you might as well apply for it."