In Developer mode, the definition of the Invoice report can be found at:

Settings --> Technical --> Actions --> Reports

• Search for the account.invoice model

• Open the Invoices Report

• Click on Qweb views

These are the views that define how the address gets printed.

Look for lines like this:

<address t-field="o.partner_id" t-options="{&quot;widget&quot;: &quot;contact&quot;, &quot;fields&quot;: [&quot;address&quot;, &quot;name&quot;], &quot;no_marker&quot;: True}"/>

Which, rewritten, really look like this:

<address t-field="o.partner_id" t-options="{"widget": "contact", "fields": ["address", "name"], "no_marker": True}"/>

Those lines use a template (called contact) to print the address. 

The template is defined in odoo/odoo/addons/base/ir/ir_qweb.xml

If you remove name from the "fields" list, then it will not be printed, and you can add something above the address line like:

 <span t-field="o.partner_id.parent_id.name"/> <span t-field="o.partner_id.name"/>

Don't forget to test for cases where one or either is blank, and to make your own view that inherits and overrides the view you want to change instead of editing the base Odoo views.