The probability, given existing evidence, is that 'The Bold Fusilier' (of which only a fragment survives; the bulk of the song is a modern composition) started out as a parody of 'Waltzing Matilda' rather than the other way around. See, for example, other discussions here where the matter has been touched upon, particularly

Craigielee/Waltzing Matilda

Marching through Rochester

There are two DT files under different titles; both correctly credit the writer (Pete Coe) but neglect to mention that he wrote his song based upon an existing fragment (one verse and chorus, first published, it appears, in The Sydney Bulletin, 1941) of uncertain origin, but of which there is no evidence prior to 'Waltzing Matilda':

COME BE A SOLDIER FOR MARLBORO [sic] AND ME

MARCHING THROUGH ROCHESTER

See also the entry at the  Traditional Ballad Index, substantially revised since an earlier, very inaccurate, version was posted (to the amusement of our Australian experts) in an old thread here.

Waltzing Matilda