Warm water and liquid soap need to be used in order to wet the stained wood. You should cover the whole painted area by running a sponge throughout the targeted area. The paint will loosen because of the water and soap since it is water-based.
The main objective is to loosen the grip of the paint on the stained wood and since it is water-based your best chances of removing it without actually damaging anything is by applying a combination of warm water and soap on it. Make sure that you take your time and use the sponge to cover all the areas with paint.
Once you feel that the paint is wet enough, grab hold of a scraper tool to chip away at the paint. You have to be very careful here as not to apply too much pressure which will damage the wood surface under the paint. The scraping process needs patience and a gentle touch.
Once you have most of the paint off, you need to take a clean sponge, soak it in water and soap and then wipe it on the remaining paint. Although sponges are soft but still it is recommended that you do not apply too much force and wipe with gentle hands.
After the paint has been wiped clean, you then need to dry the stained wood with the help of a clean dry cloth. Be sure that the fabric of the cloth is soft.
After drying check again for any paint residue and also run your hand over the wood to see if the surface is smooth. Take your time as you check for any left over paint as you will want the entire surface to be clean.