Dr. Garikipati Narasimha Rao in his discourse on Mahābhārata in this YouTube video (Telugu) says that the custom of reciting Virāṭa Parva for rains has originated because Bhīṣma describes the 18 signs of Yudhiṣṭhira's presence, in Virāṭa Parva and one of them is: "where king Yudhiṣṭhira resides...the clouds, without doubt, pour abundant rain."
In Go-harana Parva of Virāṭa Parva, this is what Bhīṣma exactly says:
The ruler, O child, of the city or the province where king Yudhishthira resides cannot have any misfortune. Charitable and liberal and humble and modest must the people be of the country where king Yudhishthira resides. Agreeable in speech, with passions under control, observant of truth, cheerful, healthy, pure in conduct, and skilful in work must the people be of the country where king Yudhishthira resides.
The people of the place, where Yudhishthira is, cannot be envious or malicious, or vain, or proud, but must all adhere to their respective duties. Indeed, in the place where Yudhishthira resides, Vedic hymns will be chanted all around, sacrifices will be performed, the last full libations will always be poured, and gifts to Brahmanas will always be in profusion. There the clouds, without doubt, pour abundant rain, and furnished with good harvest the country will ever be without fear. There the paddy will not be without grain, fruits will not be bereft of juice, floral garlands will not be without fragrance, and the conversation of men will always be full of agreeable words. There where king Yudhishthira resides, the breezes will be delicious, the meetings of men will always be friendly, and cause of fear there will be none. There kine will be plentiful, without any of them being lean-fleshed or weak, and milk and curds and butter will all be savoury and nutritious. There where king Yudhishthira resides, every kind of corn will be full of nutrition and every edible full of flavour.
There where king Yudhishthira resides, the objects of all the senses, viz.,--taste, touch, smell, and hearing, will be endued with excellent attributes. There where king Yudhishthira resides, the sights and scenes will be gladdening. And the regenerate ones of that place will be virtuous and steady in observing their respective duties. Indeed, in the country where the sons of Pandu may have taken up their abode during this thirteenth year of their exile, the people will be contented and cheerful, pure in conduct and without misery of any kind. Devoted to gods and guests and the worship of these with their whole soul, they will be fond of giving away, and filled with great energy, they will all be observant of eternal virtue.
There where king Yudhishthira resides, the people, eschewing all that is evil, will be desirous of achieving only what is good. Always observant of sacrifices and pure vows, and hating untruth in speech, the people of the place where king Yudhishthira may reside will always be desirous of obtaining what is good, auspicious and beneficial. There where Yudhishthira resides, the people will certainly be desirous of achieving what is good, and their hearts will always incline towards virtue, and their vows being agreeable they themselves are ever-engaged in the acquisition of religious merit.
In the book, Hindu Dharma: The Universal Way of Life §, Swami Chandrasekarendra Saraswati cautions these mantras and pārāyaṇa don't work always:
Even today we see how mantras are efficacious though what we see is more often their power to do evil rather than good. The very word "mantrikam" inspires dread in us. If mantras have the power to do evil, they must also have the power to do good. We do hear reports of how mantras are beneficent, for instance how the mantras invoking the god Varuna produce rains.
It may be that sometimes the Varuna japa does not succeed in bringing rains. But this is no reason why all mantras should be rejected outright as of no value. Sick people die even after the regular administration of medicine. For this reason do we condemn medical science as worthless? We have an explanation for the patient's failure to recover: his illness and reached such an advanced stage that no medicine could be of any avail. Similarly, no mantra is of any help when it has to contend against the working of powerful karma. There is also another reason. If you are not strict about your diet, the medicine taken may not work. Similarly, if we are lax in the observance of certain rules, the mantras will not produce the desired result.
[Hindu Dharma » The Vedic Religion And Varna Dharma » Preserving the Vedas: Why it is a Lifetime Mission]
§ An older and copyright-free version of this book is available for free here.