Goalkeeper Halstead, 19, and defender Eastham have put pen-to-paper to one-year deals with the option for a further year, having been regulars in the Seasider's reserves side.
Parkinson told the club's official website: "I'm delighted both players have signed professional contracts with the club.
"The majority of supporters will know all about Mark as he has appeared on the bench for the first-team on a number of occasions.
"His performances for both the reserves and youth team have attracted a lot of interest from Premier League clubs and for him to sign for the club is a real boost.
"Ash has developed into an excellent defender during the last 12 months and I'm confident that if he shows the same effort and desire that he has done during his time with us, he will make a good living out of the game."