Tenses - Reported speech WITHIN a conditional clause

If we would like to express a conditional statement about a present fact

as in "Can you go outside today? - I don't know yet, I need my parents' approval first"

Are these sentences correct?

1. If James asked his friend if she could go outside, she would tell him she needs her parents' approval first.




2. If James asked his friend if she can go outside, she would tell him she needs her parents' approval first.




3. If James asked his friend if she could go outside, she would tell him she needed her parents' approval first.

1: It sounds ok.

2: To me it is acceptable (no backshifting). However it may sound more like a general statement rather than a one-off invitation.

3: Now it's the tricky one. Couldn't it be interpreted for example as a related past event?

"If James asked his friend if she could go outside when she was 13, she would tell him that back in the day she needed her parents' approval first."

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Reported speech usually triggers a backshifting when the introductory clause is in the past. Using the same examples as in the beginning:

John asked if she could go outside today.

She told him she didn't know yet and that she needed her parents'
approval first.

What if the introductory clause is a conditional sentence?

(If he liked her) John would ask if she can go outside today

vs

(if he liked her) John would ask if she could go outside today

Answer

Your third sentence

If James asked his friend if she could go outside, she would tell him she needed her parents' approval first.

is full of unspecified and therefore ambiguous options -

1. If James asked his friend now if she could go outside now, she would tell him now that she needed her parents' approval first.




2. If James asked his friend last week if she could go outside now, she would have told him then that she needed her parents' approval first.




3. If James asked his friend last week if she could go outside last week, she would have told him last week that she needed her parents' approval first.




4. If James asked his friend now if she could go outside last week, she would tell him now that she had needed her parents' approval first.

Ambiguity is easy. The goal is clarity.