Why the text doesn't follow the parallel lines perfectly ?

If the issue is not resolved yet please provide the file. 

You are correct--Leading is the space from baseline of type to baseline of type. And your leading is off. Try putting your first line of type directly on the top line and then you will see that the lines crawl downward just a little bit. By placing your first line of type on the first line, then you can adjust the type so that every other line lines up on the succeeding line. Then once you get the leading correct, then move your type into position between the line.

If the lines themselves are slightly off, this might also account for the issue. You can always select one line of type and manually adjust the leading in the paragraph by increasing or decrerasing the leading for that one line. The keyboard shortcut is Alt-Up or Down arrow.

Michelle

Try resetting the Type tool, and then adjusting the font size rather than the leading.  Note: if you are importing existing text, then there might be an issue embedded in that. You can test that by allowing a Tpye window to auto populate after resetting, and see how it compares with the imported text.