Striker Richards, who had a spell at the club during the 2007/08 campaign, has arrived on a one-month loan from Burton.
Central midfielder Montrose has joined from Gillingham on loan until the end of the season while free-agent Parker has signed a contract until the summer.
Boss Chris Wilder told the club's official website: "Justin is a strong lad who scores his share of goals and also holds the ball up well and brings other people into the game.
"With James Constable out for the next few games we needed an experienced player up front and he fits the bill very well.
"Lewis has played over 100 games and is a very good player at this level.
"With Lee Cox and Jake Forster-Caskey returning to their parent clubs we wanted to add a midfielder to strengthen that department and we are fortunate to be able to bring in someone of Lewis' quality.
"Josh is someone we are willing to take a chance on and he deserves a chance to show what he can do over a longer period where he can get himself really fit and we can work with him."
Richards and Parker are able to play against Sheffield United in the FA Cup third round tie but Montrose is cup-tied.