First evaluate in place all subexpressions matching pattern before substituting result

In case you do have some knowledge about the expressions to be replaced and you can confidently predict that Position will not lead to nested replacements, you might still want to use a hash table to hold the intermediate results.

ReplaceAllOnceH2[expr_, rules_List] := 
 With[{p = Position[expr, Alternatives @@ rules[[All, 1]]], 
   t = System`Utilities`HashTable[]}, 
  ReplacePart[expr, 
   Thread[p -> 
     Extract[expr, p, 
      If[System`Utilities`HashTableContainsQ[t, #], 
        System`Utilities`HashTableGet[t, #], 
        With[{w = Replace[#, rules]}, 
         System`Utilities`HashTableAdd[t, #, w]; w]] &]]]]

Edit: Carl Woll's answer made me realize it should be done differently. The code below Sows the replacement values (wrapped by eval and Hold), because these may be non-injective wrt. the variables in the patterns.

eval prevents the second ReplaceAll from replacing parts that didn't match the pattern in the first place. Hold prevents evaluation until the association is constructed (whereby duplicates are deleted). First is applied to release Hold.

replaceOnce[expr_, rules_] := Module[{eval, expr2, reap},
 {expr2, {reap}} = Reap[expr /. Fold[MapAt[#2, #, {All, 2}] &, rules, {Hold, eval, Sow}]];
 expr2 /. First @@@ AssociationThread[reap -> reap]]

replaceOnce[expr, rules] // AbsoluteTiming

{2.0040347, (a + 2 b + c) (d + 2 e + f)}

Old answer:

You can use ReplaceAll to find the cases and Sow them as non-delayed rules. By putting the non-delayed rules in an Association, duplicates are automatically deleted. The association holds the evaluation, however, so Map[Identity] is applied.

Even if your rules is short, there may be many distinct matches, and if that is case replacing with an Association is usually preferable.

evaluateRules[expr_, rules_List] :=
     Module[{eval, rules2 = rules},
        rules2[[All, 1]] = eval /@ rules2[[All, 1]];
        Identity /@ (Association[Reap[expr /. #;][[2, 1]]] /. rules2) &[
            Pattern[a, #] :> Sow[a -> eval[a]] & /@ rules[[All, 1]]]]

expr /. evaluateRules[expr, rules] // AbsoluteTiming

{3.0014318, (a + 2 b + c) (d + 2 e + f)}

1.05174 seconds

This is very simple, first get the Position, then evaluate only the rules that are been used (non empty Position list ) using ParallelTable to avoid sequential evaluation. Using Table would take 2 seconds.

expr := (a parity[1] + b parity[2] + c parity[3])*(d parity[1] + 
     e parity[2] + f parity[3]);

rules = {
   parity[x_Integer?OddQ] :> (Pause[1]; 1),
   parity[x_Integer?EvenQ] :> (Pause[1]; 2),
   nonexistent[_] :> (Pause[1]; 0)
   };

LaunchKernels[]
ClearAll[replacerh];
replacerh[expr_, rules_] := With[
  {pos = Position[expr, #] & /@ rules[[All, 1]]},
  ReplacePart[expr,
   ParallelTable[
    If[
     Length[pos[[k]]] > 0,
     pos[[k]] -> rules[[k, 2]],
     Nothing
     ]
    , {k, Length[pos]}
    ]
   ]
  ]

replacerh[expr, rules] // AbsoluteTiming
(* {1.05174, (a + 2 b + c) (d + 2 e + f)} *)