Your task is to implement a TypeScript utility type called LookupPathParser.

This utility should take a string representing a lookup object path and produce an array of strings, parsing both dot notation and array notation.

Requirements:

1. Dot Notation: For paths like a.b.c.d, the result should be:

['a', 'b', 'c', 'd']

2. Array Notation: Paths that include array indices, such as a[1].b[0].c, should be parsed correctly. The result for this case should be:

['a', '1', 'b', '0', 'c']