What number Jataka tale does this story come from?

The closest I have found is about a quail who put out a fire though determination. This is VAṬṬAKA-JĀTAKA (35). Perhaps there might be another Jataka about the parrot but I am not sure if it fits the description. You can perhaps read more about other Jataka tales from The Jataka tr. by Robert Chalmers.

What the author of this had to say:

This is part of Rafe Martin, Sensei's teisho on "The Brave Little Parrot" jataka. It is from his forthcoming book on Zen Buddhist practice and the jatakas, to be published by Wisdom Publications 2016.

The text for the Brave Little Parrot Jataka remains elusive. It exists as a few stirring lines of verse in the brief Jatakastava “Verses in Praise of the Buddha’s Former Lives,” a short Sythian-Khotanese work composed just prior to 1,000 AD, discovered in the cache of texts found in the Buddhist cave grottoes of Dun-Huang, China, in 1907. It does not appear in either the Pali Jataka or the Sanskrit Jatakamala, though Jataka No. 35 in the Pali collection of 547, about a quail who stops a forest fire with an act of truth, is seen as a variant. An original may exist as a carving at the great Buddhist monument of Borobodur, as well as a painting at the Ajanta caves. Whether as text or visual art, the ending of the story is different than that of our version. In the original the god, moved by the parrot’s heroism, squeezes a cloud causing rain to fall. I created the god’s tears more than forty years ago when I began telling the tale. The colors of the parrot’s feathers aren’t mentioned either. Are they gray? Multi-colored? Never mind. The point is that deeds make us who we are. Beautiful, which means selfless deeds, make us beautiful. There is truth greater than fact. Myth reveals what never was, but always is. While fact recounts what happens in one time, in one place, myth (ideally) reveals the archetype, i.e., what happens in all times, all places. Literalists adhering to the facts can still create a truncated story.

It was my decision to make the Parrot female. Is she so in the original? No, he is not. Does that make our version untrue? Stories must find their relevance to every generation or they wither and fade. The feminine is reawakening. The jatakas are often though not exclusively, patriarchal. There are wise women, enlightened nuns, and true wives and queens throughout the tales. But rarely (if ever), does the Buddha appear in as female. Maybe this is true. Maybe this is how karma works. A man is a man is a man. Or maybe monastic forms led to suppression of a greater truth. In the jatakas, the Buddha could be a crow in a graveyard, a mongrel dog, or a monkey but never a woman? Really? I felt it was time to give our grand old tradition a nudge. Hence, the emergence peck peck peck, from its overly literal shell of the brave lady parrot. Dogen writes, in “Receiving the Marrow by Bowing,” in Shobogenzo, section no. 9:

Why are men special? Emptiness is emptiness. Four great elements are four great elements. Five skhandas are five skhandas. Women are just like that. Both men and women attain the way. You should honor attainment of the way. Do not discriminate between men and women. (Treasury of the Eye of the True Dharma: Zen Master Dogen’s Shobo Genzo, ed. by Kazuaki Tanahashi)

You may find a collection of the Jatakas given as a Dhamma-Gift by PTS in English a and by Beyerlein & Steinschulte Verlag in German in accesstoinsight format here: Jataka also a nice tool the "ramdom jataka" programmed and given by a devoted Upasaka.

There are certain "Mahayana" Versoins, which have obviously less founding in the teachings in meaning and also less connections to a serious source of origin.

It's kalandaka jatakaya, 548 jataka tale.

For those who don't understand sinhala:

The bodhisatva was born as a squirrel at one time. He had babies in his nest. Once, heavy rains and floods, removed big trees etc, and his nest was washed into the sea. To save his babies, the Bodhisatva squirrel, tried to empty the sea using his tail, by dipping it in the water, and shaking it off. After 7 days, Shakra saw his effort, and spoke to the squirrel. Shakra in human form told the squirrel that, the squirrel was doing something useless, and that he didnt understand how large the ocean was, cos he was a squirrel. But the squirrel said to him, the man himself lacked effort, and just by speaking to such a person, he the squirrel had wasted a lot of time.......... and asked the man to leave. Shakra, realising the great effort of the squirrel, helped to save the baby squirrels.

It's the short version of a beautiful story. And to think of all the pain and trouble the Bodhisattva had to go through, just to give us his teachings.