Yes, it is. However, as Vālmīki says, a hunter can kill a bird, no doubt; but only when it's alone. And not when it's with its mate both enjoying their time together.
ततः करुण वेदित्वात् अधर्मो अयम् इति द्विजः | निशांय रुदतीम् क्रौन्चीम् इदम् वचनम् अब्रैइत् || १-२-१४
Then on seeing the wailing female krounchi bird, compassion haunting him and apperceiving the killing of male bird as unjust, the sage uttered this sentence... [1-2-14]
मा निषाद प्रतिष्ठाम्त्व | मगमः शाश्वतीः समाः | यत् क्रौङ्च मिथुनात् एक | मवधीः काम मोहितम् || १-२-१५
Oh! Ill-fated Hunter, by which reason you have killed one male bird of the couple, when it is in its lustful passion, thereby you will get an ever-lasting reputation for ages to come... [1-2-15]
That makes sense.
Now fast forward to Mahābhārata.
King Pāṇḍu is in the exact same situation as the hunter from Rāmāyaṇa. Only difference is, there, it was a pair of birds and here, it's a couple of deer.
Vaisampayana said, 'O king, one day Pandu, while roaming about in the woods (on the southern slopes of the Himavat) that teemed with deer and wild animals of fierce disposition, saw a large deer, that seemed to be the leader of a herd, serving his mate. Beholding the animals, the monarch pierced them both with five of his sharp and swift arrows winged with golden feathers. O monarch, that was no deer that Pandu struck at, but a Rishi's son of great ascetic merit who was enjoying his mate in the form of a deer. Pierced by Pandu, while engaged in the act of intercourse, he fell down to the ground, uttering cries that were of a man and began to weep bitterly.
I can understand that the hunter from Rāmāyaṇa was ignorant and illiterate, therefore, Vālmīki educated him by delivering a curse. But it's rather strange that a well-educated and skilled king like Pāṇḍu wasn't aware of the rules of hunting. Had he not read Rāmāyaṇa ever? How can one explain Pāṇḍu's actions?
Pāṇḍu even defends his action saying:
It is well-known that men slay deer by various effective means without regarding whether the animals are careful or careless...
So does it mean you can't kill a bird when it's romancing with its mate, but with deer it's not a problem?
What was Pāṇḍu really doing in the forest in the first place? A couple of chapters earlier, it's said that he retired into the woods. Now why would a healthy king suddenly retire into the woods to only chase deer? Was he always fond of hunting or chasing deer?
A little while after, O bull of Bharata's race, Pandu who had achieved a victory over sloth and lethargy, accompanied by his two wives, Kunti and Madri, retired into the woods. Leaving his excellent palace with its luxurious beds, he became a permanent inhabitant of the woods, devoting the whole of his time to the chase of the deer. And fixing his abode in a delightful and hilly region overgrown with huge sala trees, on the southern slope of the Himavat mountains, he roamed about in perfect freedom. The handsome Pandu with his two wives wandered in those woods like Airavata accompanied by two she-elephants. And the dwellers in those woods, beholding the heroic Bharata prince in the company of his wives, armed with sword, arrows, and bow, clad with his beautiful armour, and skilled in all excellent weapons, regarded him as the very god wandering amongst them.
And at the command of Dhritarashtra, people were busy in supplying Pandu in his retirement with every object of pleasure and enjoyment.