Can this be simplified even further?
1k and 2k are parallel as you suggested.
You can do wye to delta transformation for 6k, 2.5k and 2k.
After this, one leg of the delta resistors will be in parallel with the 4.5k
After the third step, you can use delta to wye back.
After the fourth step, the resistor remaining in the place of 2k will be in series with the resistor you find in the first step.

There might be more simplifications. Think of wye-delta transformations whenever you are stuck.