How do I parse 天津图书馆园区馆 (tiān jīn tú shū guǎn yuán qū guǎn)?
Yes this is confusing. If you search it you'll find the (complete) name should be 天津图书馆海河教育园区图书馆. 

(天津)海河教育园区 (Tianjin Haihe Education Park) is used for specifying the location of the branch library here. Note that 园区 is usually used as a term, which means "集中发展某种事业的地区" (key development area for some kind of enterprise).

I guess that the name 天津图书馆海河教育园区图书馆 is just too long to write on the road sign so it's shorted.

At last to answer your question directly the sentence should be parsed as 


天津 = Tianjin, the city you live in.
图书馆 = library
(海河教育)园区 = (Haihe education) park 
馆 = branch library