Parallelize the construction of sparse matrices

I’m trying to parallelize a few constructions of chain complexes (given as a list of sparse matrices). I have a function that for any column (or row) computes the list of nonzero entries in that column (row), but several entries can appear at the same place $ (i,j)$ in the matrix (their values should be summed).

For example, given a simplicial complex as a list bases (simplices of given dimension), I write:

chCx[bases_] := Module[ {dim=Length@bases, dims=Length/@bases, basesk, baseskk, bdrs={}, bdr, x},    basesk =Association@Table[bases[[1,i]]->i,{i,dims[[1]]}];      Do[baseskk=Association@Table[bases[[k,i]]->i,{i,dims[[k]]}];  ... AppendTo[bdrs,bdr]; basesk=baseskk;, {k,2,dim}]; bdrs]; 

Then I replace ... with 4 different commands, to get functions chCx, chCx0, chCx1, chCx2:

bdr = SparseArray[{},Length/@{basesk,baseskk}]; Do[bdr[[basesk[Delete[s,{{r}}]],baseskk[s]]]+=(-1)^(r+1),{r,1,k},{s,Keys@baseskk}]; 


bdr=SparseArray[{},Length/@{basesk,baseskk}]; SetSharedVariable[bdr]; DistributeDefinitions[basesk,baseskk]; ParallelDo[bdr[[basesk[Delete[s,{{r}}]],baseskk[s]]]+=(-1)^(r+1), {r,1,k},{s,Keys@baseskk}]; 


bdr = ParallelCombine[ Module[{bdri=SparseArray[{},Length/@{basesk,baseskk}]},  Do[bdri[[basesk[Delete[s,{{r}}]],baseskk[s]]]+=(-1)^(r+1), {r,1,k},{s,#}]; bdri]&, Keys@baseskk, Plus]; 


bdr = SparseArray[Flatten[ParallelTable[{basesk[Delete[s,{{r}}]],baseskk[s]}->(-1)^(r+1),  {r,1,k},{s,Keys@baseskk}],1],Length/@{basesk,baseskk}]; 

When I use these commands with

n=12; bases=Table[Subsets[Range@n,{k}],{k,1,n-1}]; (*sphere*) AbsoluteTiming[chCx[bases]][[1]] AbsoluteTiming[chCx0[bases]][[1]] AbsoluteTiming[chCx1[bases]][[1]] AbsoluteTiming[chCx2[bases]][[1]] 

on a 2-core i3-560 CPU, I get:





Why is the nonparallelized code still the fastest? How can I efficiently parallelize this construction?