Josie James, from Clwt y Bont near Caernarfon, had been discharged from a mental health unit despite having "murderous thoughts towards herself".
An inquest heard her parents had not been given clear advice or an emergency phone number.
A narrative conclusion was recorded.
Miss James died in August 2015. The inquest heard on the day of her death she had told a friend she was going to jump off the bridge.
The alarm was raised immediately but she had already gone over the side.
Pathologist Dr Mark Lord said she died from a ruptured major artery due to a fall from a height and said she would have died on impact.
Miss James' mother, Joy James said: "As a family we have no interest in pointing the finger at any individual. We just want lessons to be learned."
In a statement, Betsi Cadwaladr University Health Board said: "We acknowledge that there was no effective communication with her family.
"We are working hard to improve the way we work with families, which includes ensuring they are supplied with treatment plans and additional information and advice to support the ongoing care of our service users."
It added families of patients can access its intensive community treatment team, and they are asked how they wish to be communicated with when patients are admitted.
A user group has also been set up, which the health board said helps it "listen and learn about how to make significant, meaningful improvements" in the care it provides.
Recording a narrative conclusion, assistant coroner Nicola Jones explained that she was going to move away from a suicide conclusion because it was not clear what Miss James' state of mind was at the time.
A health board has acknowledged there was "no effective communication" with the family of a 15-year-old girl who fell to her death from the Menai suspension bridge.