The visitors took control of the Division Two clash on Tuesday after two days blighted by bad weather. Essex reached 305 for 8, with Ben Foakes unbeaten on 132, and their first act on the final morning was to declare.
That left Gloucestershire, who made just 134 in their first innings, 171 adrift and facing a testing day in order to secure a draw. And it looked like defeat could well be on the cards when both openers went early but Gidman and Tavare guided them to 251 for 2 and a draw was agreed upon.
Gloucestershire lost their first wicket in the third over when Chris Dent, who was last man out in the first innings, edged behind, giving David Masters his seventh wicket of the match.
Captain Michael Klinger then fell for 7, bowled by Tymal Mills to leave the hosts wobbling at 18 for 2.
But there was nothing Essex could do to part Tavare and Gidman, who batted together for more than 70 overs, putting on 233 runs. It was the second 200-plus partnership of the match, with Foakes and James Foster putting Essex in control with 218 on the third day.
For 24-year-old Tavare, who was on exactly 100 when a draw was declared, it was a second century in three matches this season. Gidman scored at a faster rate to reach 119, his 21st first-class century.
Essex collected 11 points from the match with Gloucestershire picking up seven. Both sides have drawn two of their three matches so far this season, with Essex beating Derbyshire in their opener and Gloucestershire losing to Hampshire.