The demonstrators, led by activists not aligned to the party, wore red headbands printed with the words "Amend the 2008 Constitution".
"The current government is trying to move forward, but they can't because of the 2008 constitution," said protest organiser Pyae Phyo Zaw, who also called for elected leaders to be given oversight of the security forces.
After decades of military rule, Nobel laureate Suu Kyi took the reins in 2016 after an electoral landslide but is forced to share power with the generals.
Under the constitution drafted by the former junta, the military chief nominates a quarter of lawmakers and the ministers of defence, home affairs, and border affairs.
It also blocks Suu Kyi from becoming president, with a prohibition on presidential candidates with foreign spouses or children. Suu Kyi had two sons with her late husband, Michael Aris, a British academic.
A flyer for Wednesday's separate counter-protest called on "those who love their race and religion" to turn out to help preserve that clause.
A nationalist movement led by Buddhist monks is critical of Suu Kyi and casts the military as a protector of the Buddhist-majority nation.
A report containing thousands of amendments proposed by various political parties was submitted on Monday for debate at the parliament in the capital, Naypyitaw, but has not been made public.
Nay Phone Latt, an NLD lawmaker in Yangon's regional parliament, told Reuters one of the party's key proposals was to set a timeline for the gradual reduction of military seats in parliament, beginning with a move from 25 per cent to 15 per cent in 2021.
The NLD holds most seats in parliament, but the military lawmakers mean it lacks the 75 per cent majority needed to amend the constitution.
"We need military men’s support, so it depends on the stance of the military," Nay Phone Latt said. "But we hope that it can be accepted by the military as it would reduce bit by bit over time."
Kyaw Khine Win, another demonstrator, said he rallied in favour of amending the charter because it was written to bar Suu Kyi from leading the country and imposed "forcefully".
"We want a country which is commanded by the people," he said.
Aung San Suu Kyi’s ruling National League for Democracy (NLD) party is pushing for change despite objections from military lawmakers, who hold a veto over amendments.