Follow these steps:
Go to Google Contacts.
Sign in the account where you want to delete old contacts.
Choose contact(s) by 'check mark sign' before contacts name.
Click 'more' from above drop down menu, there you'll get delete option.
In case you have two different Google accounts added to device and one is having your old contacts then you can disable syncing of contacts of that particular account without deleting contacts.
Go to Settings and search for 'Google account' in search tab, or alternatively, open Settings → Cloud and accounts/accounts → click Google
There you will see list of all connected Google accounts. Now, simply click the account you want to disable auto syncing of any particular option, such as contacts.