How would you say: "This is my family photo taken two years ago of my younger brother's 10th birthday party."

我认为这样就可以了

这张全家福是两年前在我弟弟十岁生日聚会时拍的。

这(张)是我全家的合影，是两年前(的)弟弟(的)十岁生日时所拍的。

这两年前的全家的合影，是我弟弟(的)十岁生日时所拍的。

• Replace final particle 了 with 的. (的 serves as a pronoun particle for the object. Similar to 'which' in English)

• You should omit all the (的) in both sentences. They are not needed

Otherwise, they are grammatical, but you can rearrange the clauses in the first one to avoid using back to back 是. e.g. [It is X, it is Y] --> [It is XY] --> 这(张)是我全家两年前，弟弟十岁生日时所拍的合影照

两年前(的)弟弟(的)十岁生日时，我们照了这张照片，是我全家的合影。

• 照片 (photo) is not commonly shortened to 片.

• You should omit the (的) in this sentence too. They are not needed

Besides that, it is grammatical. You can reduce it to "两年前弟弟十岁生日时，我们照了这张全家的合影照。" to make it shorter.

合影照 short for 合影的照片