After fiddling around with this some more I figured it out.
I changed my code to this:
{{ if isset .Params "author" }}
{{ range $index, .Site.Data.authors }}
{{ if eq $.Params.author $index.author }}
<img src="{{ .thumbnail }}" alt="{{ .name }}">
<p>{{ .name }}</p>
<p>{{ .bio }}</p>
<a href="{{ .linkedin }}"></a>
<a href="{{ .twitter }}"></a>
{{ end }}
{{ end }}
{{ end }}
Now it works how it should.