What does 尽 mean and what tone does it have in 对我父母也极尽照顾?
Many dictionaries have a single entry for 极尽.
All with the pronunciation:

jíjìn

ABC

V.
use to the utmost

KEY

V
1 exhaust (one's ability, etc.), try one's best within the limits of
2 perform/fulfil to the utmost (a duty, etc.)
3 (before an adjective:) extremely, to the utmost

PLC

VERB
do one's utmost
极尽所能
jíjìn suǒnéng
do one’s best