I would not call the statement "categorically false" as /u/iamjeremybentham has, not at all. Gervase Rosser turned up cases of individual clerics owning and managing brothels in Westminster and London. Parish priests often had side jobs if their Church benefice was insufficient to support themselves (and very possibly their family); why not procuring and brothelkeeping?

The more famous case is probably the stews of Southwark. Part of Southwark lay in the legal (civic and canonical) jurisdiction of the Bishop of Winchester; part of it lay outside. The bishops legalized brothels and prostitution in their part of Southwark, but both remained illegal in the areas under secular control. What's important here is that the diocese received a cut of the profits of the brothels (both rent for the property and taxes).

And make no mistake, the bishops knew their actions contradicted what the Church was supposed to stand for.

Prostitution was made illegal in the mornings and afternoons on religious holidays.

It's categorically false.

Prohibitions against prostitution and fornication in general can be found in every edition of the Catechism ever since the council of Trent.

As to before the council of Trent, there is still no evidence of the Catholic Church operating any brothels--and you won't find any, because unlike many cannon laws, the prohibition against prostitution comes directly from the New Testament, allegedly from the mouth of Jesus himself.

Now, did Priests, Bishops, and even Popes sometimes engage in fornication, even with prostitutes? You bet. Did the Church occasionally look the other way when a brothel was running in town? Naturally, but none of this is to say that the Church actually ran one, and they especially wouldn't run one for the Priests.

Fornication and Sodomy was the currency of political attacks in the Vatican, and Alexander VI was famously accused of hosting orgies in the apostolic palace, but most people acknowledge that such events almost certainly didn't happen. If you combine internal charges to discredit enemies within the Vatican with pieces of early Protestant propaganda from without, it's not hard to see how rumors like these have endured for centuries.

There is one somewhat related issue that might be interesting to you.

Although the church formally did not approve of priests being married, many parish priests would live with a de-facto partner - although it's a bit anachronistic to call them that. I've seen "housekeeper" (with "quote marks") and "concubine" as words to describe them, although the connotations there might not give the right picture.

This is particularly the case in England, where the custom was generally to permit the marriage of priests until the 11th century reforms, where the Papacy cracked down on this (and also on issues such as simony - buying positions in the church - and of investiture - whether it's the king or the pope who gives a bishop their authority). By the time of the English Reformation, when English priests were permitted to marry again, in many cases this was simply legitimizing their existing partners as legal wives.

This is not quite what you're talking about, but I suspect the question behind your question is that you're interested in the sexual morality (and potential hypocrisy) of the church in the Middle Ages, and the fact that many priests were had "housekeepers" is relevant to that general topic.