The party will be held in the garden,weather----

weather permitting是你的句子的“独立主格”结构 == if the weather permits （it ).
又如：
If it doesn't rain tomorrow,I will take a walk in the street.
== IT not raining tomorrow,I will take a walk in the street.
要点：去掉从句部分的连接词,保留主语,把谓语动词变成分词结构（现在分词或者过去分词,看情况处理.）