ClearAll[f]
f[a_] := f[a] = Cos[a]

grid = Table[x, {x, 1, 2, 0.1}];
vals = Parallelize[f[#] & /@ grid];

In[30]:= DownValues[f]
Out[30]= {HoldPattern[f[a_]] :> (f[a] = Cos[a])}

In[31]:= grid=Table[x,{x,1,2,0.1}];
vals=f[#]&/@grid;
DownValues[f]
Out[33]= {HoldPattern[f[1.]]:>0.540302,HoldPattern[f[1.1]]:>0.453596,HoldPattern[f[1.2]]:>0.362358,HoldPattern[f[1.3]]:>0.267499,HoldPattern[f[1.4]]:>0.169967,HoldPattern[f[1.5]]:>0.0707372,HoldPattern[f[1.6]]:>-0.0291995,HoldPattern[f[1.7]]:>-0.128844,HoldPattern[f[1.8]]:>-0.227202,HoldPattern[f[1.9]]:>-0.32329,HoldPattern[f[2.]]:>-0.416147,HoldPattern[f[a_]]:>(f[a]=Cos[a])}