The Churchill Theatre recommend parking in The Hill car park (use the postcode BR2 0DA), which is a 10-15 minute walk to the theatre at buggy pace. It costs £3.60 to park here for 3 hours.
There are plenty of other car parking options in Bromley Town centre, including some which are bit closer, but please note these can be busy at peak shopping times. The Glades car park is above the main shopping centre (use the postcode BR1 1DN) and costs £3.40 for 3 hours. Please allow plenty of time to get in and out of The Glades if you're bringing a buggy and using the lift!
You can find all other parking options near the venue on the Parkopedia website.