I prefer 'political dynasty', but 'family of lawmakers' works as well.

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Note added at 1 hr (2019-04-30 19:54:22 GMT)
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Also 'political family'. See Wikipedia article:

https://en.wikipedia.org/wiki/Political_family
A political family (also referred to as political dynasty) is a family in which several members are involved in politics and businesses, particularly electoral politics. Members may be related by blood or marriage; often several generations or multiple siblings may be involved.

'Lawmakers' refers specifically to legislative service, which might be more accurate. It avoids getting stuck with the type of government: parliamentary or congressional, national or provincial/state, etc. Note how it works in the first example:

https://www.detroitnews.com/story/obituaries/2016/03/27/ex.....
Mar 27, 2016 - **Hertel comes from a family of lawmakers**. Two brothers also served in state House, Senate or U.S. House.

alt.timesofisrael.com/brazils-only-jewish-congressman-accused-of-nazi-behavior/
Jul 16, 2017 - “Sergio Zveiter's reaction filled the Jewish community with pride. **He belongs to a traditional family of lawmakers**. His father Waldemar Zveiter ...

https://www.washingtonpost.com/.../hatred-and-discrimination...
May 10, 2016 - Roman comes from a powerful family of lawmakers, a common feature in the Philippines, where political dynasties have dominated.