Holding portraits of their children who perished when the Sewol ferry sank with the loss of around 300 lives last month, they staged a sit-in at a street corner, a few hundred metres from the Blue House.
A heavy police presence, including officers in riot gear, prevented them moving any closer.
"We're calling for a meeting with the president to press our demands", a spokesman for the families, Kim Byeong-Kwon, told AFP.
They want explanations for perceived delays in the initial rescue effort, and for those responsible to be punished.
Also Read
They also want more resources deployed at the recovery site to speed up the retrieval of the bodies of around 30 passengers still unaccounted for.
Supporters of the group pasted small, yellow paper boats on the sides of police vehicles with handwritten messages reading "I'm so sorry, children", "Remember the Sewol" and "Shame on South Korea".
A few representatives of the families were later allowed into the Blue House for talks with a top presidential adviser for political affairs.
The parents then decided to march to the Blue House to demand a meeting with Park.
The Sewol was carrying 476 people when it sank on April 16 after listing sharply to one side and then rolling over.
Of those on board, 325 were children from a high school on an organised trip to the southern resort island of Jeju.
Initial investigations suggest the ferry was carrying up to three times its safe cargo capacity.