I could only find a marriage in Maughold in 1921 between William Parker, a police constable of Lancs, and Frances Pearson of Ramsey, and suggested that Lily could have been a second marriage if Frances died.
Have you tried www.manxbmd.com for any William Parkes?

Re. the illegitimate Lily last message, BL's marriage regt index has:
KERMODE, Thomas, [marr.] 9 Aug 1884, C mal 8, KELLY, Elizabeth.

The 1891 census of Patrick, Foxdale, Old Road E, has:
Thomas Kermode 35 lead miner b.IoM Patrick, with his wife Elizth 29 b.Michael, and children (all Kermode): Lily 7 b. German, Thomas 5 b.Malew, Wilfred 3 b.Patrick, Philip 2 b.Patrick, Serica E 2 mo b.Patrick, and visitor Ada Gilrea M 25 Husband at sea b.Michael.

So this Lily was the daughter of Elizabeth Kelly and Thomas Kermode, born pre-marriage. - And half-sister of Edward Clague IF the same Elizabeth Kelly (right age but wrong place of birth).

She was unmarried in the 1911 census, in a Corkill household at 2 Hawarden Avenue Douglas:
KERMODE, Lily Servant Single F 26 1885 General Servant Domestic Isle of Man Foxdale.

Sue