I’ve come across the following: "A synthetic put is created by combining a short future or stock position and a long call option trade of the same underlying. "

The above seems to ignore the bond that we consider in put call parity: p + s = c + bond

can anyone explain when we are able to exclude the consideration of a risk-free bond from a synthetic position or why the risk-free bond is necessary in the first place?

Apologies, I understand why the fiduciary call is necessary for the equation to hold. But when can the risk-free z-bond be excluded from a synthetic option position?

Take the call over to the other side
p+s-c
and look at the payoff when the options expire
The payoff from the call is max[s-X,0]
The payoff from the put is max[X-s,0]
where X is the strike price of the options
When the options expire:
If s>X then p+s-c=0+s-(s-X)=X
if s<X then p+s-c=(X-s)+s-0=X
if s=X then p+s-c=0+X-0 =X

so when the options expire p+s-c=X always
And you can also get X at expiration by holding a zero coupon bond