Do farmers have the right to kill pests?
It can not be mentioned as disallowed because it is unavoidable. But at the same time, it is recognized as a sin, and to remedy that sin, prescriptions are also being provided.
See the following verses from the Parashara Smriti's 2nd Chapter:
अयोमुखेन काष्ठेन तदेकाहेन लाङ्गली ।
पाशकोमत्स्य घाती च व्याधः शाकुनिकस्तथा ॥९॥
A hunter who makes his living by killing beasts ; one who lives by ensnaring them ; a fisherman and a fowler ; as also an agriculturist who makes no gifts (of paddy) :-— all these five incur the same identical sin.
A farmer commits sins here since because of his tilling of the soil numerous insects and worms are getting killed. Similarly, if he uses pesticide for intentionally killing the troublesome insects.
अदाता कर्षकश्चैव सर्वे ते समभागिनः ।
वृक्षं छित्त्वा महीं भित्त्वा हत्वा च कृमिकीटकान् ॥१०॥
By cutting trees, by rending the earth, and by Destroying insects and worms, what sin is incurred by a tiller of the soil is removed by the sacrifice on the threshing floor.
(Note that the sacrifice on threshing floor is nothing but making gifts of paddy at that place.)
So, as per Hinduism, the act will be a sin (although unaviodable) that needs to remedied. And, the remedy is the act of making gifts of paddy by the farmers.
Likewise, for remedying such unavoidable sins, that one does on a daily basis, in Hinduism, we have the prescription of the daily performance of the five great sacrifices.
For example, brooming a house or a place is necessary for keeping them clean. But even that act is not free from sins, because one might be killing so many unseen insects/organisms during the process. According to Hinduism even such sins are required to be remedied.
Manu Smriti 3.68. A householder has five slaughter-houses (as it were, viz.) the hearth, the grinding-stone, the broom, the pestle and mortar, the water-vessel, by using which he is bound (with the fetters of sin).
Manu Smriti 3.69. In order to successively expiate (the offences committed by means) of all these (five) the great sages have prescribed for householders the daily (performance of the five) great sacrifices.
I am not sure if the killing of insects using pesticide will be explicitly mentioned in scriptures. But the killing of several insects and worms that occur during the process of farming, is mentioned as a sin and also mentioned is the method of remedying that sin by making the paddy gift.
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