England have not included Jonny Bairstow in their Test squad for the tour of New Zealand while four players are handed maiden call-ups for the two-match series.
Batsman Jason Roy is another to miss out after a difficult Ashes series while seamer James Anderson remains unavailable due to a calf injury.
Surrey batsman Ollie Pope is back involved and Jos Buttler will be the wicketkeeper in New Zealand due to Bairstow's absence.
National selector Ed Smith confirmed Bairstow had been dropped rather than rested from the Test team but backed the Yorkshire batsman to "come back stronger" following a break.
Smith said on Sky Sports: "This feels like an opportunity for Jonny to reset.
"He does drop out of the Test squad for these two Test matches in New Zealand, which are not part of the Test Championship, and I predict he'll come back stronger having had a bit of time away from the Test set-up.
"He's not rested, he drops out of the Test team, but it's an opportunity for Jonny after a very busy summer, very hectic schedule, to reset, to work on one or two things and to come back stronger.
"My prediction is he comes back very strongly."
Smith, who is travelling to Leeds on Wednesday to speak with Bairstow about his future, assured the 29-year-old that he still has a big part to play for England.
Smith added: "He's a central cricketer for England across the formats, he's got a fantastic future in an England shirt and that is some of the things we'll be talking about face to face.
"Jonny is a very good player who has probably not been playing at his best in the last spell in Test match cricket."
Mahmood and Parkinson are also in the Twenty20 squad for the five-match series, along with Somerset wicketkeeper-batsman Tom Banton and Worcestershire seamer Pat Brown.
Banton has burst onto the scene this season, hitting 549 runs in the Vitality Blast, while Brown took 17 wickets as Worcestershire reached the final at Edgbaston last weekend.
Mahmood and Parkinson helped Lancashire finish top of the North Group before their quarter-final exit, taking eight and 21 wickets respectively.
Eoin Morgan, who led England to 50-over World Cup glory earlier this summer, will also continue to captain the T20 side.