In[0]: test0[size_]:=IdentityMatrix[size];
In[1]: Eigenvalues[test0[2]][[1]]
In[2]: ParallelEvaluate[Eigenvalues[test0[2]][[1]]]

Out[1]: 1

Out[2]: {1, 1}

In[3]: BeginPackage["testing`"];
         test1[size_]:=IdentityMatrix[size];
       EndPackage[];
In[4]: Eigenvalues[test1[2]][[1]]
In[5]: ParallelEvaluate[Eigenvalues[test1[2]][[1]]]

Out[4]: 1

Out[5]: {{{1, 0}, {0, 1}}, {{1, 0}, {0, 1}}}

In[6]: ParallelEvaluate[Eigenvalues[test1[2]]]

Out[6]: {{1, 1}, {1, 1}}