Updated Thunderbird packages are available for CentOS 5 and 6:

CESA-2015:2519 Important CentOS 5 thunderbird Security Update
CESA-2015:2519 Important CentOS 6 thunderbird Security Update

CESA-2015:2519 Important CentOS 5 thunderbird Security Update

CentOS Errata and Security Advisory 2015:2519 Important

Upstream details at : https://rhn.redhat.com/errata/RHSA-2015-2519.html

The following updated files have been uploaded and are currently
syncing to the mirrors: ( sha256sum Filename )

i386:
bb8e25af5ff6287b1e25e1f022e4fa5194721bf442d1ad84523cc3000b320d8e thunderbird-38.4.0-1.el5.centos.i386.rpm

x86_64:
b234caf73d4d5b8819dc8ba3fd8a15f9a221b8f90dd3e4ebdbb9331dd5dcf86c thunderbird-38.4.0-1.el5.centos.x86_64.rpm

Source:
080d0b3b6404131402cc2da5db51316ba96c6944488f8d44a7d788cdcf82a24d thunderbird-38.4.0-1.el5.centos.src.rpm

CESA-2015:2519 Important CentOS 6 thunderbird Security Update

CentOS Errata and Security Advisory 2015:2519 Important

Upstream details at : https://rhn.redhat.com/errata/RHSA-2015-2519.html

The following updated files have been uploaded and are currently
syncing to the mirrors: ( sha256sum Filename )

i386:
7d94acf6d46a4ffa415d8eefece825154c79f6c080b694c3f0780dd9b65271d6 thunderbird-38.4.0-1.el6.centos.i686.rpm

x86_64:
655c0162a6bb9d918c8a497531329d5ba13225a2cea9ce5f46756426ad066cbc thunderbird-38.4.0-1.el6.centos.x86_64.rpm

Source:
58473f1a2f8951dd8f8b2276f07e3c2da8894a1f8ebaa65b0b4d90d6eb63a50b thunderbird-38.4.0-1.el6.centos.src.rpm