Python is dug into the default CentOS 6 install.  You can’t just remove it.  Unfortunately you also can’t upgrade it using the system “yum upgrade” routine because CentOS doesn’t publish any updates for it.  The solution is pretty easy.  This was all referenced on this website:

I am republishing the commands taken from that site here.  All credit should go to the author of the above link.

# Start by making sure your system is up-to-date:
yum update
# Compilers and related tools:
yum groupinstall -y “development tools”
# Libraries needed during compilation to enable all features of Python:
yum install -y zlib-devel bzip2-devel openssl-devel ncurses-devel sqlite-devel readline-devel tk-devel gdbm-devel db4-devel libpcap-devel xz-devel expat-devel
# If you are on a clean “minimal” install of CentOS you also need the wget tool:
yum install -y wget
# Python 2.7.14:
tar xf Python-2.7.14.tar.xz
cd Python-2.7.14
./configure –prefix=/usr/local –enable-unicode=ucs4 –enable-shared LDFLAGS=”-Wl,-rpath /usr/local/lib”
make && make altinstall