Are children of Brahmin parents also Brahmin by caste?
Yes, caste is based on birth. There is scriptural support for it.
Manusmriti (10.5).- In all castes (varna) those (children) only which are begotten in the direct order on wedded wives, equal (in caste and married as) virgins, are to be considered as belonging to the same caste (as their fathers)
Āpastamba (2.13.1).—‘Sons begotten by a man who approaches in the proper season a woman of equal caste, who has not belonged to another man, and who has been married legally, have a right to follow the occupations of their castes.’
Viṣṇu (16.1).—‘On women equal in caste to their husbands, sons are begotten who are equal in caste to their fathers.’
Yājñavalkya (1.90).—‘From women of the same caste as their husbands are born sons of the same caste.’
Baudhāyana (1.17.2).—‘Sons of equal caste spring from women of equal caste.’
The genetic makeup of the 4 castes are different. Brahmanas have Nadis in their body situated in such a way as to receive the vision of the Lord. Kshatriyas are genetically strong and physically fit. Vaishyas have a mind and body conducive for conducting business. Shudras are naturally pessimistic, lazy, simple-minded, and materialistic. That's how their bodies were constructed.
Source for Nadis in a Brahmana's body situated in a particular way. At 39:00 minutes in the video.
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