If the parent of a text frame is a character, then it is an anchored text frame. Thus, the "parent paragraph", as you put it is:
If the frame is on the pasteboard as a free-standing page item, then its parent is the spread rather than a page. If it is on there because of anchoring, then you have to work harder because the parental chain will give you the page that holds the text frame that holds the character to which the frame is anchored. You'll need to use geometric bounds in that case.