Parsimony is the principle generally used in science that "if there is no evidence to the contrary, it is best to assume the simplest explanation is true". This assumption is arguably one of the cornerstones of the scientific method. In phylogenetics, "parsimony" is also commonly used to refer to one of the commonly-used methods for inferring phylogeny, which is based on this principle - see maximum parsimony for more on this.
Unfortunately, a great deal of confusion has arisen from misunderstanding of this principle - in particular, from the common failure to realise that parsimony is simply a working assumption, rather than a statement of fact. Parsimony seems to rest on the basic requirement that the universe is inherently simple, which many would regard as doubtful at best. However, the advantage of applying parsimony to scientific analysis is that it is generally a limiting factor. There is usually only one simplest explanation, whereas if we are willing to accept more complicated suggestions then there may be no limit to how far we can go. Running with the simplest option therefore allows us to phrase our explanation in the manner most likely to be falsifiable, which allows it to be further tested rigorously. Such further testing may, of course, lead to the discovery that the explanation fails to account for all the data, and must be abandoned in favour of a different explanation - this is the basis of doing science.
A hypothesis or theory which involves fewer unproved assumptions is other things being equal more likely to be true. Each time a new assumption is added there is the possibility that the new assumption is false. Therefore theories based on few assumptions or based on probable assumptions are more likely to be true.