But, in the Season 1 finale, “Auld Lang Syne,” it’s revealed that the funeral viewers have been seeing throughout is actually for Kate’s dad, Bud Mularkey (Paul McGillion).

Plus, the reason why Tully hasn’t been present leading up to the funeral is because she’s estranged from Kate at this particular time. She ultimately does show up to the funeral (proving that she’s not the one who has died), and Kate tells her that nobody wants her there.  

Source: https://www.distractify.com/p/firefly-lane-does-tully-die