If (button.interactable) condition doesn't work - why?

The check is only made once: When this script first awakens it checks to see if the button is enabled, if it is, the entire typewriter effect will occur. If the button is not enabled, then the typewriter effect will never occur (unless the script itself is disabled and re-enabled).

If you want to wait for the button to become enabled, you need to use the Update() method. If you want to stop the typewriter effect after it starts, you'd have to restructure your code entirely.

using System.Collections;
using System.Collections.Generic;
using UnityEngine;
using UnityEngine.UI;

public class UITextTypeWriter : MonoBehaviour {

    public Text txt;
    private string story;
    public Button ParrentButton;
    private bool writing;

    void Awake()
    {
        txt = GetComponent<Text>();
        story = txt.text;
        txt.text = "";
        writing = false;
    }

    private void Update()
    {
        if (!writing)
        {
            Button b = ParrentButton.GetComponent<Button>();
            if (b.interactable)
            {
                writing = true;
                StartCoroutine("PlayText");
            }
        }
    }

    IEnumerator PlayText()
    {
        foreach (char c in story)
        {
            txt.text += c;
            yield return new WaitForSeconds(0.125f);
        }
    }
}

The writing boolean is so we don't check the button again after we've started (or the coroutine would run more often than we want).