rest easy for the balance of their days

[Quang Hai]

I am reading a short story. A Reverend wants to stop his younger brother to tell a story to his parent.

'Don’t distress your mother and your father'
He paused, then spoke with emphasis, and quietly.
‘Your mother and father wouldn’t rest easy for the balance of their days.’

I do not fully understand underlined words.thanks!

[Gillnetter]

I am reading a short story. A Reverend wants to stop his younger brother to tell a story to his parent.

'Don’t distress your mother and your father'
He paused, then spoke with emphasis, and quietly.
‘Your mother and father wouldn’t rest easy for the balance of their days.’
For the remaining days of their lives. For the rest of their lives. From the time that you distress your mother and father until the time that they die.

I do not fully understand underlined words.thanks!

Gil