One has the finite flat group scheme $\mathbb Z/p$ over $\mathcal O_{K_{\mathfrak p}}$
(I write $K_{\mathfrak p}$ for the $\mathfrak p$-adic completion of $K$, and 
$\mathcal O_{K_{\mathfrak p}}$ for its integer ring), 
as well as
the finite flat group scheme $A[p]$.  Giving a $p$-torsion point over $K_{\mathfrak p}$ (and hence in particular over $K$) is the same as giving a closed embedding
on generic fibres:
$(\mathbb Z/p)\_{/ K_{\mathfrak p}} \hookrightarrow A[p]\_{/K_{\mathfrak p}}.$

Raynaud's results imply that this extends to a closed embedding over $\mathcal O_K$:
$\mathbb Z/p \hookrightarrow A[p],$
which is another way of saying the that the non-zero $p$-torsion point has non-zero
reduction.

Just to see concretely what can happen in the situation when $e \geq p-1$, suppose
that $K = \mathbb Q$ and $p = 2$.  Then we could have a map
$(\mathbb Z/2)\_{/\mathbb Q_2} \hookrightarrow A[2]_{/\mathbb Q_2}$
which extends to a closed immersion
$\mu_2 \hookrightarrow A[2].$
This would correspond to having a 2-torsion point in the kernel of the reduction map.
(Note that $\mu_2$ has a non-trivial point in char. zero, which collapses down to
the identity in char. two.)