In March the weather is still rather cold. It can even snow. Also, Paris gets 24 mm of rainfall during 14 days, so you'll probably see a shower or some light rain during your stay. Moderate rain is the most typical precipitation for March.
In April the weather gradually improves. The temperature rises from 9 °C to 12.5 °C by the end of the month which makes it possible to sit outside in a cafe or go for a picnic. With an average of 25 mm of rainfall during 14 days you will also definitely get wet.
May is the wettest and warmest spring month which already feels like summer. In recent years, the highest temperature in Paris reached 32°C. This month also brings some thunderstorms, rains get more abundant. With an average of 26 mm of rainfall during 12 days you'll definitely need an umbrella.