First, let us generate a large sum with terms similar to your mini-example.
hugeExpr[n_] :=
Plus @@ Array[
RandomInteger[{-10, 10}] x^
RandomInteger[{-3, 1000}] Sqrt[x + RandomInteger[{-10, 10}]]^
RandomInteger[{-1, 1}] &, n];
expression = hugeExpr[300000];
The actual number of addends (Length[expression]) might not be 300K, but it will still be a large count. Now, the answer to your problem.
ParallelMap[Series[#, {x, 0, 2}] &, expression, Method -> "ItemsPerEvaluation" -> 15]
I tried it on a expression with over 40K terms and it took less than 2 seconds. If your expression is orders of magnitude larger than that, I would suggest you split it in chunks of say 100K addends, use ParallelMap on each chunk separately, save those results to individual files, and add them up later.