I have a sub-domain which has many errors like duplicate titles, description, 404 errors, ... but the parent domain is working perfectly.
I want to know if the sub-domain errors effects the parent domain SEO?
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Sub-domains can positively effect the parent domain just as they can negatively effect the parent domain.
Minor errors are not a big issue. You only have to fix them. Generally, smaller errors such as this speaks to content quality. It is important that your content is high quality and performs well. Otherwise, particularly low quality content of any related site including sub-domains will effect your other sites. I always advise that as new sites are introduced to the web, that they be as high quality as is possible even if development is not complete. Make sure what you offer is good and not poor.
In your case, if you only have minor errors, fix them relatively quickly and do not worry about performance issues. Minor issues that do not exist for too long will not effect your parent domain or any other site.
I will list some answers to give you an idea of how this works.
This link explains that sub-domains receive some of their metrics from the parent domain and how a parent domain and sub-domain are related (explained only in part).
The following links refer to potential penalties due to links from sub-domains, parent domains, and related sites. While this is not your scenario, it may be useful to read in order to understand that sub-domains, parent domains, and any other related domains are linked. It is important to realize that while sub-domains must perform in the SERPs as a unique site, they are far from divorced from any parent domain.
No, subdomains are considered to be seperate sites entirely.