After a lot of back and forth and reading various posts on this I thought I would submit my answer. There maybe better ways of doing what I ended up doing but hopefully this will provide a start for someone wanting to do the same.
First here are a few posts and questions that helped:
My Solution
First I created a search form that had the checkboxes for the search, each checkbox name when posted would form the get variable for the filter. A check is done to see if there are categories, if there are then I loop through and add them. There is also a check done to see if you are on the filter results page, if you are then set the relevant checkboxes to checked on the form.
The Search Form Code
<form action="{{ url('homes-for-sale/filter') }}" method="get">
{# check if there are any park categories if there are then show them as checkboxes #}
{% if parkCategories|length %}
{% for category in parkCategories %}
{% set loopIndex = 'park' ~ loop.index %}
<label class="checkbox-inline">
{# if you are on the filter results page then set the results to checked #}
{% if template is defined and template == 'filter' %}
<input type="checkbox" name="{{ loopIndex }}" value="{{ category.slug }}"{% if category.slug == park1 or category.slug == park2 or category.slug == park3 or category.slug == park4 or category.slug == park5 %} checked{% endif %}>
{% else %}
<input type="checkbox" name="{{ loopIndex }}" value="{{ category.slug }}">
{% endif %}
{{ category.navTitle }}
</label>
{% endfor %}
{% endif %}
{# check if there are any price categories if there are then show them as checkboxes #}
{% if priceCategories|length %}
{% for category in priceCategories %}
{% set loopIndex = 'price' ~ loop.index %}
<label class="checkbox-inline">
{# if you are on the filter results page then set the results to checked #}
{% if template is defined and template == 'filter' %}
<input type="checkbox" name="{{ loopIndex }}" value="{{ category.slug }}"{% if category.slug == price1 or category.slug == price2 or category.slug == price3 or category.slug == price4 or category.slug == price5 or category.slug == price6 %} checked{% endif %}>
{% else %}
<input type="checkbox" name="{{ loopIndex }}" value="{{ category.slug }}">
{% endif %}
{{ category.title }}
</label>
{% endfor %}
{% endif %}
<button type="submit">Submit</button>
</form>
The Filter Results Page
On the results page you need to check which filter variables have been set via the get request and then display the appropriate related entries.
{# set park search parameters #}
{% set park1 = craft.request.getParam('park1') %}
{% set park2 = craft.request.getParam('park2') %}
{% set park3 = craft.request.getParam('park3') %}
{% set park4 = craft.request.getParam('park4') %}
{% set park5 = craft.request.getParam('park5') %}
{# set price search parameters #}
{% set price1 = craft.request.getParam('price1') %}
{% set price2 = craft.request.getParam('price2') %}
{% set price3 = craft.request.getParam('price3') %}
{% set price4 = craft.request.getParam('price4') %}
{% set price5 = craft.request.getParam('price5') %}
{% set price6 = craft.request.getParam('price6') %}
{# set park category model if it is in the search query #}
{% set parkCategory1 = park1|length ? craft.categories.slug(park1).first().id %}
{% set parkCategory2 = park2|length ? craft.categories.slug(park2).first().id %}
{% set parkCategory3 = park3|length ? craft.categories.slug(park3).first().id %}
{% set parkCategory4 = park4|length ? craft.categories.slug(park4).first().id %}
{% set parkCategory5 = park5|length ? craft.categories.slug(park5).first().id %}
{# create an array of park categories #}
{% set parks = [
parkCategory1,
parkCategory2,
parkCategory3,
parkCategory4,
parkCategory5,
] %}
{# create a park array and trim blank strings #}
{% set trimParks = [] %}
{% for park in parks %}
{% if park|length %}
{% set trimParks = trimParks|merge({ (loop.index0): park}) %}
{% endif %}
{% endfor %}
{# set price category model if it is in the search query #}
{% set priceCategory1 = price1|length ? craft.categories.slug(price1).first().id %}
{% set priceCategory2 = price2|length ? craft.categories.slug(price2).first().id %}
{% set priceCategory3 = price3|length ? craft.categories.slug(price3).first().id %}
{% set priceCategory4 = price4|length ? craft.categories.slug(price4).first().id %}
{% set priceCategory5 = price5|length ? craft.categories.slug(price5).first().id %}
{% set priceCategory6 = price6|length ? craft.categories.slug(price6).first().id %}
{# create an array of price categories #}
{% set prices = [
priceCategory1,
priceCategory2,
priceCategory3,
priceCategory4,
priceCategory5,
priceCategory6,
] %}
{# create a price array and trim blank strings #}
{% set trimPrices = [] %}
{% for price in prices %}
{% if price|length %}
{% set trimPrices = trimPrices|merge({ (loop.index0): price}) %}
{% endif %}
{% endfor %}
{% if trimParks|length and trimPrices|length %}
{% set entries = craft.entries.section('homesForSale').relatedTo('and',
{targetElement:parks},
{targetElement:prices}
) %}
{% endif %}
{% if trimParks|length and not trimPrices|length %}
{% set entries = craft.entries.section('homesForSale').relatedTo('or',
{targetElement:parks}
) %}
{% endif %}
{% if trimPrices|length and not trimParks|length %}
{% set entries = craft.entries.section('homesForSale').relatedTo('or',
{targetElement:prices}
) %}
{% endif %}
<h1>Filter Results</h1>
{% if entries|length %}
<p>{{ entries|length }} results:</p>
<ul>
{% for entry in entries %}
{% set park = entry.park.first() %}
{% set price = entry.parkPriceCategory.first() %}
<li><a href="{{ entry.url }}">{{ entry.title }} :: <b>{{ park }}</b> :: <b>{{ price }}</b></a></li>
{% endfor %}
</ul>
{% else %}
<p>Your search didn't return any results.</p>
{% endif %}
