Can a relationship exist that is mutualistic/parasitic/etc. but not considered symbiotic?
TL;DR: mutualistic/parasitic relationships are members of symbiotic relationships class

So, first of all, there is a way to look it up.

Secondly, the very same article in wiki has subsection on parasitism. Reading it would be useful:


  This is also known as antagonistic or antipathetic symbiosis.


Also, brief search through Google Scholar will show how these words are used in context. papers include such as:


Fungal symbiosis from mutualism to parasitism: who controls the outcome, host or invader?
Mutualism and parasitism: the yin and yang of plant symbioses
Mutualism or parasitism? The variable outcome of cleaning symbioses
Enhanced growth as a manifestation of parasitism and shell deposition in parasitized mollusks. Aspects of the biology of symbiosis.