Double publication & parallel submitting a same study as a Paper and Patent
In other words, is a published patent besides its copy as journal paper considered as double publication?


Typically there is no copyright problem, because there is no overlap in the text.  A patent consists of formal legal writing, and it generally does not use any of the text of the academic paper.  If you wanted to use some of the same text, then you would need to make sure the publisher of the paper did not object.  However, this is uncommon: academic and legal writing styles differ substantially, and they are aiming at very different purposes, so any substantial overlap would indicate that you are doing a poor job of at least one of the two (and you might as well avoid minor overlap so that you don't have to address the copyright issue).

Even aside from copyright, there is still the question of whether this is considered a form of double publication or simultaneous submission.  I believe the answer is almost always no: patents don't count as academic publications and publishers will not care or object.  However, it can't hurt to ask before submitting, just to make sure this is how it works in your field and for this particular journal.  It's also a good idea to mention the patent submission in your cover letter when submitting the paper, so that nobody could accuse you of concealing this information.  (I'd be shocked if it were a problem, but better safe than sorry.)

Keep in mind that very little of the academic content will actually appear in the patent application.  Academic papers are typically focused on experiments, data analysis, or theory, but these would not be included in the patent application itself.  Instead, it is a description of an invention and how it is used in practice, together with a long list of legal claims laying out which aspects of the invention are worthy of legal protection.  Academics could learn something from the description of the invention, but it in no way substitutes for the academic paper.