Ashley Randele came on "TMZ Live" Monday to tell her tale -- saying her father, Tom Randele, came clean with an estimated 6 weeks to live following a lung cancer diagnosis ... suddenly off-loading the decades-long secret as they watched NCIS.

The 71-year-old explained to Ashley and her mom that he had to change his real name ... and they ought to finally know in case it came up after his passing ... because authorities "were probably still looking for him."

Ashley says she pressed him for more info the next day after realizing it wasn't a "terrible dad joke" -- adding it was vital for her to know her dad's real name: Ted Conrad.

Ignoring his pleas not to look into it, Ashley later found herself searching his name online ... yielding various "vault teller robs bank" headlines ... and recent articles stating the 50-year hunt was still active.

The revelation left her stunned ... after all, she had just learned the man she always perceived as an average suburban dad stole $215,000 --- the equivalent of $1.8M today -- while working as a vault teller at the Society National Bank in Cleveland, Ohio, on Friday, July 11, 1969, as a 20-year-old college dropout.

He fled to Washington, D.C., that night ... and when bank bosses discovered the theft on Monday, the FBI was called, and Conrad became a wanted man.

He ended up in Boston ... assuming a new identity and resuming normalcy despite his picture plastered across headlines.

Whether the heist was worth it or not would have been a question for Ted, AKA, Tom ... as Ashley says, most of the stolen money was blown on ritzy digs and at least one bad investment ... reaching the point of her loaning her parents 10K before they filed for bankruptcy in 2014.

Ashley goes in depth about her story on her podcast, "Smoke Screen: My Fugitive Dad."