Since mother was ill in the hospital I had to care for her.

连词since 表示“从...以来至今” 意义时，主句谓语动词使用现在完成时。网友的句子，是与从句相同的一般过去时。

所以，since 表示原因。此句只有根据时态判断since 的意思。

如果主句是完成时 have had，那么since 表示“时间”：

Since mother was ill in the hospital I have had to care for her.

自从妈妈生病住院以来，我不得不照顾她。

又如：

Since his illness, he has had to stay out of tennis. 

自从生病以来，他不得不一直远离网球了。