It was Jefferson who was a part of the Democrat-Republican party. Adams was a federalist and his letter to Jefferson reveals an animosity that he held, not against Jefferson himself, but in a general sense, against the Democrats of his day. If it were not so, he would have made no mention of them and therefore must have had reason to list them amongst the parties he hated.
Do your homework in depth.
John Adams was Pro-Administration (before 1795), Federalist (1795 – 1808) and a Democratic-Republican (from 1808) until his death. The correspondence between him and Jefferson (of which the linked letter is one) was after they had both left government. The letters were to settle the feud between them.
The reason why John Adams had become a Democratic Republican is because the federalist party no longer existed after the defeat of John Adams' bid for a second Presidential term. The defeat that the federalists had suffered was so devastating that they ceased to exist as a viable political party and as a consequence, the country was essentially either under one-party rule or at least dominated by one party for several decades until the formation of the Republican Party shortly before the Civil War. After the dissolution of the federalist party, John Adams simple had no choice but to become a Democratic-Republican, but then again, whether they liked it or not, neither did anyone else. Until the rise of the Whigs and free-soiler parties---the free-soilers being an abolitionist party---no one had a choice as to what political party to align themselves with.
As for the final letters between Jefferson and Adams, there is a very interesting story behind what began the resumption of their friendship as the reconciliation between Adams and Jefferson had been inspired by a dream or vision given to their mutual friend, Benjamin Rush:
Do your homework in depth.
John Adams was Pro-Administration (before 1795), Federalist (1795 – 1808) and a Democratic-Republican (from 1808) until his death. The correspondence between him and Jefferson (of which the linked letter is one) was after they had both left government. The letters were to settle the feud between them.
The reason why John Adams had become a Democratic Republican is because the federalist party no longer existed after the defeat of John Adams' bid for a second Presidential term. The defeat that the federalists had suffered was so devastating that they ceased to exist as a viable political party and as a consequence, the country was essentially either under one-party rule or at least dominated by one party for several decades until the formation of the Republican Party shortly before the Civil War. After the dissolution of the federalist party, John Adams simple had no choice but to become a Democratic-Republican, but then again, whether they liked it or not, neither did anyone else. Until the rise of the Whigs and free-soiler parties---the free-soilers being an abolitionist party---no one had a choice as to what political party to align themselves with.
As for the final letters between Jefferson and Adams, there is a very interesting story behind what began the resumption of their friendship as the reconciliation between Adams and Jefferson had been inspired by a dream or vision given to their mutual friend, Benjamin Rush:
To John Adams From Benjamin Rush October 17, 1809
https://founders.archives.gov/?q=Correspondent%3A%22Adams%2C%20John%22%20Correspondent%3A%22Rush%2C%20Benjamin%22&s=1111311111&r=147
From John Adams To Benjamin Rush October 25, 1809
https://founders.archives.gov/?q=Correspondent%3A%22Adams%2C%20John%22%20Correspondent%3A%22Rush%2C%20Benjamin%22&s=1111311111&r=148
To John Adams From Benjamin Rush December 5, 1809
https://founders.archives.gov/?q=Correspondent%3A%22Adams%2C%20John%22%20Correspondent%3A%22Rush%2C%20Benjamin%22&s=1111311111&r=149
From John Adams To Benjamin Rush December 21, 1809
https://founders.archives.gov/?q=Correspondent%3A%22Adams%2C%20John%22%20Correspondent%3A%22Rush%2C%20Benjamin%22&s=1111311111&r=150