I am reproducing the question part:
Dasharatha killed a brahmin (unintentionally) and didn't leave his
kingdom for forest.
Pandu killed brahmin and left kingdom.
Rama killed Ravana, a brahmin, and didn't do anything as parihara.
- What are the parihara for sins of brahmin killing?
Let me answer point-wise:
- Dasaratha never killed any brAhmin by accident. The ascetic boy, who was killed by Prince Dasaratha, was born to a an ascetic couple - Husband being a Vyshya and wife being a Sudra.
न द्विजातिर् अहम् राजन् मा भूत् ते मनसो व्यथा | शूद्रायाम् अस्मि
वैश्येन जातः जन पद अधिप || २-६३-५३
'O, king the ruler of the country! I am not a Brahmana. Let there be
no agony in your mind. I am born through a Sudra woman by a Vysya.
- Pandu did not abdicate his kingdom just because he killed a sage by mistake. He was overcome with grief and decided to roam in the forest like an ascetic.
'The wicked, even if born in virtuous families, deluded by their own
passions, become overwhelmed with misery as the fruit of their own
deeds. I have heard that my father, though begotten by Santanu of
virtuous soul, was cut off while still a youth, only because he had
become a slave to his lust. In the soil of that lustful king, the
illustrious Rishi Krishna-Dwaipayana himself, of truthful speech,
begot me. A son though I am of such a being, with my wicked heart
wedded to vice, I am yet leading a wandering life in the woods in the
chase of the deer. Oh, the very gods have forsaken me! I shall seek
salvation now. The great impediments to salvation are the desire to
beget children, and other concerns of the world.
I shall now adopt the Brahmacharya mode of life and follow in the
imperishable wake of my father.
I shall certainly bring my passions under complete control by severe
ascetic penances. Forsaking my wives and other relatives and shaving
my head, alone shall I wander over the earth, begging for my
subsistence from each of these trees standing here.
- Sri Rama did not kill a brAhmana, ie., Ravana. Ravana declared himself as a kshatriya, a person from warrior class.
While sending Indrajit to defeat Sri Hanuman, Ravana says as follows:
न खल्वियम् मतिः श्रेष्ठा यत्त्वाम् सम्प्रेषयाम्यहम् | इयम् च
राजधर्माणाम् क्षत्रियस्य मतिर्मता || ५-४८-१३
"This thought of my sending you is not indeed the best option. But,
this decision is said to be according to the duties of a king and is
the commendation of a warrior-class."
Agni Purana says:
A killer of a brahmana has to build a hut in the forest and live there
for twelve years. He has to beg for a living and give up all that he
possess to another brahmana.