MPs on Wednesday voted 73 to 13 in favour of the proposal, which has already been supported by the canton's government.
Businesses in the canton are spearheading efforts to hold the sporting event, just two years after voters rejected a plan to host the 2022 Winter Games in Davos and St. Moritz.
The bid is largely supported by right-wing parties but opposed by the Socialists, the SDA news agency reported.
Graubünden is not the only Swiss canton seeking to host the 2026 Games.
The government of Valais has backed a plan by businessman Christian Constantin, who is also president of the Sion football club.
The Winter Olympics have been staged twice in Switzerland in 1928 and 1948 — both times in St. Moritz.