Sorry, messed up the formatting above. Here's the error again:

An Unexpected Error occurred while upgrading: #0 /www/fortyone/public_html/updates/concrete5.6.2.1_updater/concrete/libraries/3rdparty/adodb/adodb.inc.php(1074): adodb_throw('mysqlt', 'EXECUTE', 1142, 'INDEX command d...', 'DROP INDEX cPar...', false, Object(ADODB_mysqlt)) #1 /www/fortyone/public_html/updates/concrete5.6.2.1_updater/concrete/libraries/3rdparty/adodb/adodb.inc.php(1049): ADOConnection->_Execute('DROP INDEX cPar...', false) #2 /www/fortyone/public_html/updates/concrete5.6.2.1_updater/concrete/libraries/3rdparty/adodb/adodb-datadict.inc.php(378): ADOConnection->Execute('DROP INDEX cPar...') #3 /www/fortyone/public_html/updates/concrete5.6.2.1_updater/concrete/libraries/3rdparty/adodb/adodb-xmlschema03.inc.php(1737): ADODB_DataDict->ExecuteSQLArray(Array, false) #4 /www/fortyone/public_html/updates/concrete5.6.2.1_updater/concrete/core/models/package.php(206): adoSchema->ExecuteSchema() #5 /www/fortyone/public_html/updates/concrete5.6.2.1_updater/concrete/core/controllers/single_pages/upgrade.php(226): Concrete5_Model_Package::installDB('/ww...') #6 /www/fortyone/public_html/updates/concrete5.6.2.1_updater/concrete/core/controllers/single_pages/upgrade.php(247): Concrete5_Controller_Upgrade->refreshDatabaseTables(Array) #7 /www/fortyone/public_html/updates/concrete5.6.2.1_updater/concrete/core/controllers/single_pages/upgrade.php(44): Concrete5_Controller_Upgrade->do_upgrade() #8 [internal function]: Concrete5_Controller_Upgrade->view('tools', 'required', 'upgrade') #9 /www/fortyone/public_html/updates/concrete5.6.2.1_updater/concrete/core/libraries/controller.php(264): call_user_func_array(Array, Array) #10 /www/fortyone/public_html/updates/concrete5.6.2.1_updater/concrete/core/libraries/controller.php(234): Concrete5_Library_Controller->runTask('view', Array) #11 /www/fortyone/public_html/updates/concrete5.6.2.1_updater/concrete/core/libraries/view.php(783): Concrete5_Library_Controller->setupAndRun() #12 /www/fortyone/public_html/updates/concrete5.6.2.1_updater/concrete/tools/upgrade.php(6): Concrete5_Library_View->render('/upgrade') #13 /www/fortyone/public_html/updates/concrete5.6.2.1_updater/concrete/startup/tools_upgrade_check.php(9): include('/ww...') #14 /www/fortyone/public_html/updates/concrete5.6.2.1_updater/concrete/dispatcher.php(105): require('/ww...') #15 /www/fortyone/public_html/concrete/startup/updated_core_check.php(6): require('/ww...') #16 /www/fortyone/public_html/concrete/dispatcher.php(21): require('/ww...') #17 /www/fortyone/public_html/index.php(2): require('/ww...') #18 {main}

Same problem here:

An Unexpected Error occurred while upgrading: #0 L:\Websites\heartbeatcorvette\concrete\libraries\3rdparty\adodb\adodb.inc.php(1074): adodb_throw('mysqlt', 'EXECUTE', 1142, 'INDEX command d...', 'DROP INDEX cPar...', false, Object(ADODB_mysqlt)) #1 L:\Websites\heartbeatcorvette\concrete\libraries\3rdparty\adodb\adodb.inc.php(1049): ADOConnection->_Execute('DROP INDEX cPar...', false) #2 L:\Websites\heartbeatcorvette\concrete\libraries\3rdparty\adodb\adodb-datadict.inc.php(378): ADOConnection->Execute('DROP INDEX cPar...') #3 L:\Websites\heartbeatcorvette\concrete\libraries\3rdparty\adodb\adodb-xmlschema03.inc.php(1737): ADODB_DataDict->ExecuteSQLArray(Array, false) #4 L:\Websites\heartbeatcorvette\concrete\core\models\package.php(206): adoSchema->ExecuteSchema() #5 L:\Websites\heartbeatcorvette\concrete\core\controllers\single_pages\upgrade.php(226): Concrete5_Model_Package::installDB('L:/Websites/hea...') #6 L:\Websites\heartbeatcorvette\concrete\core\controllers\single_pages\upgrade.php(247): Concrete5_Controller_Upgrade->refreshDatabaseTables(Array) #7 L:\Websites\heartbeatcorvette\concrete\core\controllers\single_pages\upgrade.php(59): Concrete5_Controller_Upgrade->do_upgrade() #8 [internal function]: Concrete5_Controller_Upgrade->view('tools', 'required', 'upgrade.php') #9 L:\Websites\heartbeatcorvette\concrete\core\libraries\controller.php(264): call_user_func_array(Array, Array) #10 L:\Websites\heartbeatcorvette\concrete\core\libraries\controller.php(234): Concrete5_Library_Controller->runTask('view', Array) #11 L:\Websites\heartbeatcorvette\concrete\core\libraries\view.php(783): Concrete5_Library_Controller->setupAndRun() #12 L:\Websites\heartbeatcorvette\concrete\tools\upgrade.php(6): Concrete5_Library_View->render('/upgrade') #13 L:\Websites\heartbeatcorvette\concrete\startup\tools_upgrade_check.php(9): include('L:\Websites\hea...') #14 L:\Websites\heartbeatcorvette\concrete\dispatcher.php(105): require('L:\Websites\hea...') #15 L:\Websites\heartbeatcorvette\index.php(2): require('L:\Websites\hea...') #16 {main}

Tried upgrading from 5.6.1 and 5.6.1.2 to 5.6.2 or 5.6.2.1.

Tried it through the dashboard, and also using method 2 of the manual upgrade described here:
http://www.concrete5.org/documentation/how-tos/developers/manually-...

Same result every time...

I'd never have thought to google that part. Thank you :D

Sorry but i didn't understand what did you do. I have the same issue. Could you explain pls? You suggest to give index privilege to the database user?

You have to execute a mysql command.
first enter mysql by typing at the command prompt :
mysql -p -u root
then enter your mysql root password.
then at the mysql prompt enter :
GRANT INDEX ON concretedb.* TO 'concreteuser'@'localhost' ;

concretedb : is the name of your concrete database (DB_DATABASE in your /concrete/config/site.php)
concreteuser : is the database user name (DB_USERNAME in your /concrete/config/site.php)
localhost : leave this if mysql is on the same server (DB_SERVER in your /concrete/config/site.php)