The gunman, who later got into a shootout with police in the building, also was in critical condition after shooting himself, said Dallas police spokesman, Sr. Cpl. Lt. Kevin Janse.
Police said the gunman and the father and son apparently had an ongoing dispute, but it was not clear why the suspect opened fire on them on the third floor of the Four Forest building in northern Dallas. The father, 63, and his 39-year-old son were in stable condition at a nearby hospital.
Witnesses said blood gushed from both sides of the son's neck as he made his way down an escalator after the shootings and pleaded for help.
"He was screaming and crying," said Abraham Achar, a witness who was visiting his friend at the United Texas Bank on the first floor. "He said, 'He shot my dad."'
Becky Hayes, who works in a first-floor office near the security desk, said she recognized the son because he sometimes ate lunch in the building cafeteria. She said she convinced him to sit in a chair until help arrived.
"He kept telling me: 'Stop the bleeding. Stop the bleeding,"' Hayes said. "I was just trying to keep him calm."
During the shootings, about 16 people took refuge in vaults in the bank.
"The building has announced we should stay in here until they let us know otherwise," bank employee Linda Farley told WFAA-TV of Dallas and Fort Worth by telephone from the vault. "Everybody is a little scared."
Police did not immediately release more information including the identities of the gunman and the father and son.