Although the parliament doesn't have the power to split up Google (or any company for that matter), this dramatic move would put political pressure on the European Union to double-down on the antitrust investigation that it has been conducting against Google for the last four years.
European concern about Google stems from the fact that the company has huge market share in Europe — about 90%, versus around 68% in the US. Some believe that it abuses that dominance by downgrading the search results of its rivals.
When the antitrust investigation first launched in 2010, now-former VP of the European Commission Joaquin Almunia listed four main areas of concern:
That Google gives links to its own "vertical search services," like Google Shopping, restaurant reviews, news, or YouTube, preference over rival links
That it takes content from competing companies (like restaurant reviews from Yelp) and uses it in its own services
That it shuts out search advertising competitors on websites where it delivers search advertisements
That it makes it difficult for advertisers to move their advertising away from its own system, AdWords.