What does the scene, in "Phase the Second: Maiden No More, XIV," between Tess and the village parson, and its aftermath, reveal about Thomas Hardy,...

This question presupposes that Hardy intrudes into the text through the ideas or voice of the third person narrator. While in some works of literature, this may be a faulty presupposition, it happens that it is correct in this case: Hardy is intruding into the text through the thoughts and voice of the narrator. As a result, it is fairly easy to analyze the narrator's remarks to learn what ideas and convictions Hardy himself holds.

The scene you mention occurs immediately after Tess's infant unexpectedly dies and her father has refused to allow her to call the village parson to christen the unnamed infant and to perform death rites for the babe. Durbeyfield has locked the door and taken the key with him to bed. When the parson calls at the house, having heard of the baby's illness, the door is locked and no one allows him to enter. This is important later when Tess has her conversation with the parson because he is annoyed that he was rejected by the household. The "aftermath" you mention is the infant's burial in the dark of night in a remote and unhallowed "shabby corner" of the cemetery where "unbaptized infants" were laid out with diverse sinners.

the strange tenderness in her voice, combined to affect his nobler impulses—or rather those that he had left in him after ten years of endeavour to graft technical belief on actual scepticism.

This quote spoken by the narrator subtly reveals that Hardy has no confidence--one might say no belief--in clergymen's belief in Scriptural precepts they teach and uphold. He is understood as thinking that ecclesiastic's religion is "technical belief" rather than true religion and true spirituality. In other words, Hardy can be understood as thinking clergymen are unthinking, unbelieving, skeptical hypocrites whose religious training destroys their human and noble feelings and impulses.

So the baby was carried in a small deal box, ... and buried by lantern-light, ... in that shabby corner of God's allotment where He lets the nettles grow, and where all unbaptized infants, notorious drunkards, suicides, and others ... are laid.

This statement by the narrator reveals that Hardy blames the Christian God (or the Christian idea of their God) for the unmerciful events that happen to the unfortunate in life, such as the laying of an innocent babe next to social outcasts thus confirming the baby as an outcast as well. It can be extrapolated from this that Hardy was not an adherent of the Christian religion as it was conceived and practiced in his culture.

If you examine the rest of the intrusive narrator's remarks in this scene, you may be able to identify other traits of Hardy's beliefs or personality. For instance, you might think about what the narrator's ironic wit reveals about Hardy: "[a job] had been unskillfully botched by his customers." You might also think about what Hardy's allusion to the philosophy of humanism reveals about him or about his ideas of the clergy: "The man and the ecclesiastic fought within him,...."