I actually just solved this problem myself. There is indeed a method of pywikipediabot that does this - it's called templatesWithParams()
Here is a code example where I am looking to get see what the "VIAF" parameter is of the {{Authority Control}} template:
def determineAuthorityControlTemplate(nameOfPage):
"""returns 'noACtemplate' if no Authority Control Template, 'templateNoVIAF' if AC template but no VIAF number,
and returns the viaf number if it exists"""
namepage = Page(enwp,nameOfPage)
templates = namepage.templatesWithParams() #here we load the templates into a list
for template in templates: # go through the templates
if template[0] == 'Authority control': # are we looking at the right template?
for param in template[1]: #go through the paramaters of the template
if param[:4] == 'VIAF': #are we looking at the right parameter?
return param[5:]
return 'templateNoVIAF'
return 'noACtemplate'