To solve this, we first need to convert all the mixed numbers to improper fractions:

15 3/19 = (19 15) + 3 = 285 + 3 = 288/19
6 5/19 = (19 6) + 5 = 114 + 5 = 119/19
2 14/19 = (19 * 2) + 14 = 38 + 14 = 52/19

Now we can substitute these fractions back into the original expression:

288/19 - (119/19 + 52/19)
= 288/19 - (171/19)
= 288/19 - 171/19
= (288 - 171)/19
= 117/19

Therefore, 15 3/19 - (6 5/19 + 2 14/19) = 117/19.