It's insane we are nearing this picture's 30 years anniversary! I was trying to find this exact information a couple of years ago, and it's a shame that so little is known about this iconic image, and the raw data are nowhere to be found. The closest I could get to a time taken was from various news articles from back then that I tracked online, reporting on the scheduled photographs to be taken (JPL news article, UPI article, New York Times article). Assuming the articles are accurate, about 64 photos were to be taken starting at around 1990-02-14 01:00 UTC and for a duration of 4 hours (in actuality they were 60 total if the caption for the original PIA00452 image is to be believed, but this caption has other inaccuracies, see below). The UPI article has a more exact-looking time reported, so going with that, the pale blue dot photo was taken somewhere around 1990-02-14 01:12 UTC and 05:12 UTC. Interesting fact: Although the caption in the PIA00452 image says that "Earth was a crescent only 0.12 pixel in size", according to my Celestia simulation, earth should appear as slightly gibbous, with a phase angle @ approx. 82.5 deg., not crescent from the position of Voyager 1 during that time (see attached rendered image).