This question has been answered by Bruno17. See the first response.
&ids=`[[!getReqParam? &name=`parent`]]`
I just need more of an explanation about Install getReqParam and the listbox settings.
<form class="sisea-search-form" action="[[~[[+landing:default=`[[*id]]`]]]]" method="[[+method:default=`get`]]">
<fieldset>
<select name="parent" id="parent">
<option value="0">Search entire site</option>
<option value="2">Cat. 1</option>
<option value="19">Cat. 2</option>
<option value="5">Cat. 3</option>
</select>
<label for="[[+searchIndex]]">[[%sisea.search? &namespace=`sisea` &topic=`default`]]</label>
<input type="text" name="[[+searchIndex]]" id="[[+searchIndex]]" value="[[+searchValue]]" placeholder="Search">
<input type="hidden" name="id" value="[[+landing:default=[[*id]]]]">
<input type="submit" value="[[%sisea.search? &namespace=`sisea` &topic=`default`]]" />
</fieldset>
</form>[[!getResources?
&ids=`[[!getReqParam? &name=`parent`]]`
]][[!SimpleSearchForm? &tpl=`Search Form` &landing=`17` &ids=`[[getReqParam? &name=`parent`]]` ]]
[[!SimpleSearch?
&ids=`[[!getReqParam? &name=`parent`]]`
]]
<select name="parent" id="parent">
<option value="0">Search entire site</option>
<option value="2">Cat. 1</option>
<option value="19">Cat. 2</option>
<option value="5">Cat. 3</option>
</select>
<select name="parent" id="parent">
<option value="0">Search entire site</option>
[[!getResources?
&parents=`0`
&tpl=`tplSearchOption`
]]
</select><option [[!getReqParam:is=`[[+id]]`:then=`selected="selected"`:else=``? &name=`parent`]] value="[[+id]]">[[+pagetitle]]</option>