Another approach that is supported by the previous versions (11.3 at least) is to create a list of functions which are either (Identity which doesn't touch the data or fn which apply your function) and then MapThread this list with Construct:

Block[{},

(* create a list of Identity function *)
temp = ConstantArray[Identity, Length[L]];

(* replace the positions with your function *)
temp[[B]] = #^2 &;

(* apply your list of function to your data (element-wise) *)
t2 = MapThread[Construct, {temp, L}];

] // MaxMemoryUsed // AbsoluteTiming

(* Out: {0.0161858, 800568} *)

Result:

t2 == SubsetMap[#^2 &, L, {#} & /@ B]
(* Out: True *)

Another benefit is the memory footprint (could vary):

MapAt[#^2 &, L, {#} & /@ B]; // MaxMemoryUsed // AbsoluteTiming
(* Out: {1.2336, 1912832} *)

(* @Syed answer *)
SubsetMap[#^2 &, L, {#} & /@ B]; // MaxMemoryUsed // AbsoluteTiming
(* Out: {0.0305638, 6541432} *)

which is around 2 and 8 times smaller respectively.

Notes:

• Minimum version 11.3 is because of Construct, if you happen to have an alternative, the minimum requirement would be lowered (although performance would be affected).
• This method could easily be extended to apply multiple functions.

Another approach that is supported by the previous versions (11.3 at least) is to create a list of functions which are either (Identity which doesn't touch the data or fn which apply your function) and then MapThread this list with Construct:

Block[{temp},

(* create a list of Identity function *)
temp = ConstantArray[Identity, Length[L]];

(* replace the positions with your function *)
temp[[B]] = #^2 &;

(* apply your list of function to your data (element-wise) *)
result = MapThread[Construct, {temp, L}];

] // MaxMemoryUsed // AbsoluteTiming

(* Out: {0.0161858, 800568} *)

Result:

result == SubsetMap[#^2 &, L, {#} & /@ B]
(* Out: True *)

Another benefit is the memory footprint (could vary):

MapAt[#^2 &, L, {#} & /@ B]; // MaxMemoryUsed // AbsoluteTiming
(* Out: {1.2336, 1912832} *)

(* @Syed answer *)
SubsetMap[#^2 &, L, {#} & /@ B]; // MaxMemoryUsed // AbsoluteTiming
(* Out: {0.0305638, 6541432} *)

which is around 2 and 8 times smaller respectively.

Notes:

• Minimum version 11.3 is because of Construct, if you happen to have an alternative, the minimum requirement would be lowered (although performance would be affected).
• This method could easily be extended to apply multiple functions.