No. Wood and its finish can react to prolonged water, heat and detergent differently from ordinary dishwasher-safe crockery. Follow the utensil maker's instructions.
Understanding this situation
A wooden handle on an otherwise suitable pan creates the same need to check the complete item. A short or cool programme is not universal permission, and damage may affect joints or finish rather than cleaning.
A useful next step
Use the specified alternative care where required. If wood warps or splits after washing, document the item's instructions before diagnosing an appliance fault. Do not continue processing deteriorating pieces that may release fragments.
Manuals and supporting guidance
Model-specific instructions take priority. These references support the practical principles discussed here; they do not diagnose your appliance.