The normal word for this overlook.

Ignore: to refuse to pay attention to; disregard.

From the Latin ignōrāre: in- ‘not’ + gno-, a base meaning ‘know’. It implies the refusal to take notice of or acknowledge; disregard intentionally.

Disregard:

to pay no attention or heed to; ignore; to show no evidence of attention concerning (something); to treat without proper respect or attentiveness. - AHDEL

To overlook, to ignore, to disregard, to neglect should be appropriate to use in place of this idiom. The meaning of the idiom is to ignore deliberately, to pretend not to notice. (See the idiom's entries in WordReference.com and CollinsDictionary.com.) In fact there are many synonyms for this.

According to Phrases.org, the origin of the idiom goes like this: Admiral Horatio Nelson is supposed to have said this when wilfully disobeying a signal to withdraw during a naval engagement during the battle of Copenhagen in 1801, when the Admiral Sir Hyde Parker sent a signal (by use of flags) for Nelson to disengage. Nelson was convinced he could win if he persisted and that's when he said to have 'turned a blind eye' on the flag signals. Chapter VII of the biography The Life of Horatio Lord Nelson by Robert Southey claims the actual words of Nelson to Captain Thomas Foley were:

"... You know, Foley," turning to the captain, "I have only one eye,—I have a right to be blind sometimes:" and then putting the glass to his blind eye, in that mood of mind which sports with bitterness, he exclaimed, "I really do not see the signal!"

And thus the idiom "to turn a blind eye" on something/someone.

Thus, using ignore, dismiss, disregard, overlook, pretend not to see should be fine depending on the exact requirement of the sentence. For example:

1. Only parents could overlook that kind of behavior.

   Only parents could turn a blind eye on such behavior.

2. Teachers overlooked the bad handwriting in cases where students were able to secure only passing marks.

   Teachers turned a blind eye to the handwriting in cases where students were able to secure only passing marks.

3. The corrupt inspector agreed to pretend not to see the safety violations.

   The corrupt inspector agreed to turn a blind eye to the safety violations.

4. The Management ignores the bullying at the workplace.

   The Management turns a blind eye on the bullying at the workplace.

Neglected will be suitable word in this context. According to Cambridge Dictionary Neglect is defined as "to not give enough care or attention to people or things that are your responsibility".

Obviously there are lots of great answers above, but I thought I'd add a fun one that I always liked even though it is archaic. It is the word "wink". If, for example, you "wink" at someone's transgression you are closing your eyes to it. It is an archaic word but I always liked the mind picture it evoked, sufficient that even those unfamiliar with the word's particular meaning, it would nonetheless convey the sense.

"Since the writing was so dynamic the reviewer often winked at his sloppy grammar so that he could enjoy the story."

Wiktionary: Wink - (archaic, intransitive) To turn a blind eye.

disengage

to stop being involved with a person or group : to stop taking part in something

Used as an intransitive verb, it is more specifically defined as:

to release or detach oneself

close one's eyes to (something)

To ignore something, to pretend that something is not really happening

The man closed his eyes to the problems that were happening in his company. The Idiom Connection

Connive at/in carries the same meaning as turn a blind eye or deaf ear. It means to secretly allow (something considered immoral, illegal, wrong, or harmful) to occur: you have it in your power to connive at my escape. From Latin connivere ‘shut the eyes (to),’ from con- ‘together’ + an unrecorded word related to nictare ‘to wink.’

in the most egregious cases, complicit or "be complicit"

involved with others in an illegal activity or wrongdoing (Oxford)

When a wrong-doing or injustice is committed and others see this but do nothing, they are essentially complicit via inaction, even if they didn't explicitly help the offender execute the crime.