Here is a solution with 12 primes. Cells with brackets have to be pressed as often as their number indicates. Other cells will not be pressed. Note that cells have to be pressed in a checkerboard pattern. Dotted cells contain even, non-prime numbers. I used a small program to find primes that are sums of three other primes. I am not sure whether 12 is optimal.

 [ 3]  31  [23] ....
  37  [ 5] .... [19]
 [29] .... [13]  43
 .... [17]  41  [11]
 

Here is a solution with 12 primes. Cells with brackets have to be pressed as often as their number indicates. Other cells will not be pressed. Note that cells have to be pressed in a checkerboard pattern. Dotted cells contain even, non-prime numbers. I used a small program to find primes that are sums of three other primes.

 [ 3]  31  [23] ....
  37  [ 5] .... [19]
 [29] .... [13]  43
 .... [17]  41  [11]
 

With the idea of JLee, we can upgrade this to 13 primes:

  [ 3]  31  [23] ....
  .... [ 5] .... [29]
  [ 2]  37  [13]  53
   19  [17]  41  [11]