The answer is no. Take the "classical" example of the line with two origins. This space is non-Hausdorff, paracompact and doesn't admit partitions of unity.
EDTI: I think the question is a kind of "duplicate":
http://mathoverflow.net/questions/19219/smooth-manifolds-that-dont-admit-a-partition-of-unityduplicate" . Ok, but if you have an example for a non-Hausdorff manifold, which doesn't admit partitions of unity, you have an example for a non-Hausdorff paracompact space with the same property.
First the definition:
The line with two origins is the quotient space of two copies of the real line
$\mathbb{R} \times {a}$ and $\mathbb{R} \times {b}$.
with equivalence relation given by
$(x,a) \sim (x,b)\text{ if }x \neq 0$.
Since all neighbourhoods of $0_a$ intersect all neighbourhoods of $0_b$, it is non-Hausdorff.
However, this space is paracompact, since $\mathbb{R}$ is paracompact.
For the non-existence of a partition of unitiy: take the open covering $ U = (-\infty,0) \cup { 0_a } \cup (0,\infty)$ and $\tilde{U} = (-\infty,0) \cup { 0_b } \cup (0,\infty)$. Assume, there is a partition of unity subordinate to this cover. Then the value of each origin would have to be $1$ which cannot be true. (Edit: villemoes was a little faster :-) )
