This is how I suggest doing it. Literally it means "being appealed (against)", that is to say, being the object of an appeal, and the subject is of course the "auto" denying the request. But this typically Spanish way of expressing it, using the gerund, is awkward in English. Moreover, as I shall explain, I don't think "appeal", the normal translation of "recurrir" in this legal context, is the best word here.

First, then, I suggest using a main verb rather than a gerund; I think this makes it clearer and more fluent.

Second, why "petitioned"? Because of what follows: "Reforma y Subsidiario de Apelación". This means that the party whose request had been denied lodged both a "recurso de reforma" and a "recurso de apelación". The former is an appeal to the court that made the ruling; the latter is to a higher court. Only the latter really corresponds to what we would call an appeal in British or American courts. The former, "recurso de reforma", is not unlike an interlocutory appeal, but I think it would be better to express it as I have suggested, "petitioned the court to revise its decision/ruling". The idea is that you have to try a "recurso de reforma" first, then if that's turned down, you appeal to the higher court with a "recurso de apelación", "subsidiariamente", which you might perhaps translate as "secondarily", though it appears that, as often happens, the two types of "recurso" were actually lodged at the same time. It goes on to say that the "recurso de reforma" was denied, so the "recurso de apelación", the appeal to a higher court, would then have gone ahead.

So I would do it something like this: "by a ruling of [date] the request made in this letter rogatory by the Judicial Authorities of XXX was refused. The court was petitioned to revise this ruling and a secondary appeal to a higher court was lodged ["filed" in American English] on [date] by XXX, and the petition for revision was denied by a decision of [date]."

All this is based on an extremely useful answer to a previous question on the phrase "en reforma y subsidiariamente en apelacion", which is well worth consulting:
http://esl.proz.com/kudoz/spanish_to_english/law_general/430...