First, one should note that poems are not essays and do not necessarily have central "ideas"; often they have dominant moods or themes, but there is nothing inherently ideological or argumentative in the genre.
The poem is probably autobiographical. Lord Bryon, himself notoriously promiscuous (with both men and women and even his half sister), had an affair with a married woman (actually the wife of one of his friends), Lady Frances Wedderburn Webster. After the affair ended, Lady Frances took up with an even more famous and equally promiscuous man, the Duke of Wellington.
The main theme of the poem is the regret and sorrow the narrator feels about the end of his relationship with the woman in the poem, described only as "you." The poem suggests that the woman may well have been the one to break off the affair:
Pale grew thy cheek and cold,
Colder thy kiss...
Now, the woman appears to have become notorious in some way (the affair with Wellington, probably, although the poem does not make it explicit). The reference to her "broken vows" suggests scandal (as she was a married woman having an indiscreet affair). As he hears her name being discussed, he is revisited by the sorrow he felt at the end of their relationship and also feels empathy for the shame of her affair being "outed" in the newspapers and her being the subject of gossip. He then speculates about what it would be like to encounter her in the future and suggests he would meet her "with silence and tears."
The main theme seems to be that the complicated emotions of regret associated with the ends of relationships are made even sadder and more complex in the case of illicit relationships.
Lord Byron's romantic poem "When We Parted Two" deals with the theme of forbidden love. The first stanza introduces the two lovers, parting in "silence and tears" (2). As the poem continues, the reader can conclude that the parting of the two lovers was on unhappy terms, based on some "deception" of the lover mentioned in the last stanza. The speaker of the poem concludes that if they should meet again, he should only greet her with "silence and tears," echoing the phrase which described their parting.
Byron would certainly understand about forbidden love. This poem is most likely about his love affair with Lady Frances Webster. Not only was she a married woman, but she was also the wife of one of his friends. Later after their affair ended, Byron learned of Lady Frances' new affair with the Duke of Wellington, which could be the reason why the speaker of the poem claims:
"In silence I grieve
That thy heart could forget
Thy spirit deceive" (26-28).
The speaker of the poem is clearly upset that his lover has parted from him and found a new conquest, as indicated by the line "thy heart could forget." If she remembered their forbidden love as well as he, she would not have found a new conquest.
Lord Byron's poem "When We Two Parted" resonates with the bitterness of an ended love affair.