Also the use of For loop is very awkward and can be much simplified for readability:

data = ParallelTable[
   pf = \[Rho][r];
   pf /= Tr[pf];
   ef = Chop[Eigenvalues[pf]];
   fdiag = 0;
   fSp = 0;
   Do[
    If[Chop[pf[[i, i]]] =!= 0,
     fdiag -= pf[[i, i]] Log2[pf[[i, i]]]
     ];
    If[ef[[i]] =!= 0,
     fSp -= ef[[i]] Log2[ef[[i]]]
     ];
    ,
    {i, 1, Length[\[Rho][r]]}
    ];
   (fdiag - Re[fSp])/4
   ,
   {r, 0.01, 3.5, 0.01}
   ];

But even these loops are not really needed as the entire thing can be handled in a vector manner:

ClearAll[NewLog2]
SetAttributes[NewLog2, Listable];
NewLog2[val_] := If[Chop[val] == 0, 0, Log2[val]]
data = ParallelTable[
   pf = \[Rho][r];
   pf /= Tr[pf];
   ef = Chop[Eigenvalues[pf]];
   fdiag = -Total[Diagonal[pf] NewLog2[Diagonal[pf]]];
   fSp = -Total[ef NewLog2[ef]];
   (fdiag - Re[fSp])/4
   ,
   {r, 0.01, 3.5, 0.01}
   ];