There is no universal answer, but there is usually a clearly better answer for a given family once you look at cash flow, health, and how showable the home is.
Selling after the move: the common choice
Moving first means your parent never lives through showings, the home can be staged properly, and repairs happen without disruption. It requires covering community costs for one to three months before proceeds arrive.
Most communities accept a deposit and a start date, and many families bridge the gap with savings, a short term loan, or a family contribution repaid at closing.
Selling first: when it is the right call
If proceeds are required for the deposit, or the home would sit empty and uninsured through winter, selling first is the safer path. Build in a rent back or a temporary stay with family so nobody moves twice.
Questions that decide it
How many months of double cost can the budget absorb? Is the health situation stable? Will an empty house be watched through a Colorado winter? Does the community hold apartments, and for how long?