"He is senex, but is he pater? We're not told of any children, and his habit of hanging out with hermits and sailors does not lead our thoughts in the direction of family life. Maybe that's one of the points of the poem, written (after all) soon after Coleridge himself became a father. In the marginal prose glosses Coleridge added to the poem, the white-bird albatross is described as ‘the pious bird of good omen’. Pietas is a particular Roman virtue associated not just with duty, righteousness and obedience to divine command, but with duty to one's family in particular. It's not just a piety but a filial piety. Might it be that killing the pious bird is impious in a specifically familial sense?
The Romans had several sacred birds, including eagles and doves, but the specific phrase pious bird, pia avis refers specifically to the Stork and crosses over from Roman use to Christian (Saint Ambrose thought the stork embodied pietas) because of its supposed devotion to its young: ‘ad pedes Ciconia apparet, quia preces fundit publicas & privatas; vel ob piétatem erga parentes: quam ob causam hanc avem piam vocant Romani’ [Eucharius Gottlieb Rink, De veteris numismatis potentia et qualitate lucubratio (1701), 171].
In all countries where the Stork breeds it is protected; boxes are provided on the tops of the houses; and he considers himself a fortunate man whose roof the Stork selects. There is a well authenticated account of the devotion of a Stork, which, at the burning of the town of Delft, after repeated and unsuccessful attempts to carry off her young, chose rather to remain and perish with them than leave them to their fate. Well might the Romans call it pia avis! [Henry Baker Tristram, The Natural History of the Bible (1867) 248]
The stork's association with childbirth survives into popular belief even today. Conceivably, the Mariner's killing of this pia avis is a cutting-off of his own potential as a father and progenitor."