How do I refer to a regex group inside a custom grok pattern?

How can I say use capture group 1 or tell grok to use my regex capture group?

By default, only named capturing groups are considered by grok, numbered capturing groups do not trigger a field creation. If you want to override this behavior, set named_captures_only to false:

named_captures_only
- Value type is boolean
- Default value is true
If true, only store named captures from grok.

However, there is nothing wrong in using a named capturing group (and I'd use a negated character class [^&]* instead of a lazy matching dot with a consuming & after it):

\bparam1=(?<param1>[^&]*)

[^&]* matches 0 or more characters other than &, and thus will also match the empty parameter (that you may want to avoid by changing * to +, or control with the keep_empty_captures parameter) and at the end of the string.