The answer is yes. All the green colored parts of the plants contains chlorophyll and they are able to carry out photosynthesis. However, most plants have more than one pigment. They may have carotenoids (that appear as yellow/orange), anthocyanins (that appear as red/purple), chlorophyll (that appears as green and carry out photosynthesis). One of these pigments will be dominant and that is the colour that the plant part will show. Even in leaves that are all red in colour chlorophyll is still present, it's just that there are higher amounts of anthocyanins in them which make them look red. As they have chlorophyll in them they can carry out photosynthesis. Hence, colourful plants can also carry out photosynthesis.