Mathematica automatically simplifies b*c*b*a to a*b^2*c but you did not tell Simplify that b^2==0 or that the product of a subscripted variable times the square of a subscripted variable equals zero.

Add one more rule for squares

Simplify[Subscript[A,sn]*Subscript[A,sp]*Subscript[A,sn]*Subscript[d,11], 
   {Subscript[A,x_]*Subscript[A,y_]==0,Subscript[A,z_]^2==0}]

returns

0

Now think if there is a*b*a*c*a == a^3*b*c

Simplify[Subscript[A,sn]*Subscript[A,sp]*Subscript[A,sn]*Subscript[A,sq]*Subscript[A,sn], 
   {Subscript[A,x_]*Subscript[A,y_]==0,Subscript[A,z_]^2==0}]

returns

Subscript[A,sn]^3*Subscript[A,sp]*Subscript[A,sq]

because there is no rule for simplifying cube powers.

Perhaps you could study the documentation and think how to write a rule that says any positive power of a subscripted variable will equal zero. Look up Condition in the documentation and see if you can think how to use that to say that any positive power of a Subscript will equal zero.

Please test that carefully to see that it works as you expect.

Try

Simplify[Subscript[H,SS]*Subscript[H,KK],Subscript[H,x_]*Subscript[H,y_]==0]

which returns

0

because x_ matches something named x while y_ matches something different named y