Not my area of expertise but I had had the exact opposite understanding, that the Democratic Party traditionally claims Jefferson as their own with the Jefferson-Jackson Day Dinners, the parallel of which for the GOP are the Lincoln Day Dinners.

Well I'm not sure Republicans do claim Jefferson and I think the Democratic claim is a big stretch at best. The simplest answer is that very broadly speaking the party that Jefferson and Madison founded in the early to mid 1790's was called the Republican party, this party split into factions in the 1820's that would become the core of the Whig and Democrat parties. Historians sometimes refer to them as Jeffersonian Republicans or Democratic-Republicans in an effort to differentiate between Jefferson and Lincoln's parties. With that said it should be noted that took some time for the parties of Jefferson and Hamilton to really coalesce around one name, in the mid 1790's elections there were some 30+ different name variations running from Federalist-Republicans to Republican-Federalists sometimes even in the same state! As a social revolution sweeps the United States and calls for universal white male suffrage increased Republicans, particularly in the north, often embraced the term democratic to further differentiate themselves from there more socially stratified federalist brothers. Federalists generally didn't acquiesce to the Jeffersonians "owning" the terms Republican and democratic, with Hamilton musing after the election of 1800 that it may not be too late for Federalists to win back the terms.

Regarding Federalist and Anti-Federalists, these were broad factions organized and coordinated almost solely on the state level not proto-parties like the Republicans and Federalists. The Anti-Federalists attempted to win the first post-ratification elections in an attempt to make the new government moot, but were unsuccessful. By the end of Washington's first term differences over policy had caused the beginnings of the two parties with most of the anti-federalists joining one of the respective factions for a variety of reasons.

TO circle things back up Republicans may claim Jefferson because he has the same party name, or because most parties in the 19th century that emerged claimed their lineage from Jefferson and company

I agree with /u/CptBuck. Do you have an example of the GOP claiming Jefferson?