I have been to Paris many times...

[alrod]

A conversation:

- Did mom tell you? I've been to Paris!
- Oh? I've been to Paris many times...Lovely city!

Can second phrase be in past simple tense? like:

- Oh? I was in Paris many times...

Is it noticeable mistake (if at all)?

[Casiopea]

Welcome, alrod.

The simple past is possible, yes. The second speaker adopts the present perfect because it mirrors or agrees with what the 1st speaker said.

All the best.

[alrod]

thank you.......

[Casiopea]

You're welcome.

[alrod]

One more question(s)
Can the perfect tense be used in first sentence? Like:

- Has mom told you I had been to Paris?

It looks right grammatically (I think) but I hear something awkward in that.

or may be

- Has mom told you I was in Paris?

It may be right too but that little changes the meaning (I was in Paris but may be long time ago, not recently) does it?

[Casiopea]

Take a look here first: Noun Clauses/Reported Speech (Indirect Speech)

[alrod]

Great!
So I correct myself:

- Has mom told you that I was in Paris?
- Did mom tell you that I had been to Paris?

(the last it a little heavy, my opinion)

[Casiopea]

Why does it sound 'heavy' to you?

[alrod]

Because I learned simple tenses before perfect and instinctively I prefer to use simple past instead present/past perfect.
So, personally, I would say:

- Did mom tell you that I was in Paris last week?

(simple past with pointing to the specific time)

[Casiopea]

Continuity helps too: Did...was...

All the best.