Left-back Meredith has been ruled out long term after breaking a bone in his foot, while winger Reid sustained a season-ending knee injury at Sheffield United at the weekend when he tore his anterior cruciate ligament.
"To lose him and Mez clearly isn't ideal," Parkinson told the club's official website. "They have formed such a great partnership for us down that left-hand side, but we have to take it on the chin and look to bounce back.
"It is clear those two have worked well together for us over the last 18 months or so. We've got to try and rectify that and make sure we are as effective down that side again.
"We need to look to address the holes in our squad caused by these injuries, especially down our left-hand side.
"We are looking to recruit some more bodies into the squad, but it isn't easy.
"There are many things to factor in, not just finances. We are trying though."
Injuries have been steadily mounting up for Parkinson this season. Teenage striker Ollie McBurnie, winger Raffaele De Vita, plus midfield pair Ricky Ravenhill and Nathan Doyle all missed Saturday's derby draw at Sheffield United through injury, while defender Andrew Davies (knee) has been out since early October.