<xsl:template name="urlParentTemplate">
  <xsl:param name="parent"/>
  <xsl:value-of select="concat('*',$parent/YMIA:Id)"/>
  <xsl:if test="ancestor::$parent[2]"> <!-- it gives error here -->

if parent is a node the line that you show would not cause an error, but
teh default value of parent used if you do not call thi stemplate using
xsl:with-param is the empty string <xsl:param name="parent"/>
but it looks like you don't want a template at all, why all these named
templates?

the grandparent of the current node is ../.. and it looks like you want
somthing like

<xsl:apply-templates select="../.."/>

David