Javascript, 140 132 Bytes

Thanks to @KevinCruijssen for -8 Bytes!

c=>{n=[];l="";for(p of(c+',').split`,`){m=l.split(/[([{]/).length==l.split(/[)\]}]/).length;m&&l?n.push(l):0;l=m?p:l+','+p}return n}

TIO

Javascript, 140 132 114 Bytes

Thanks to @KevinCruijssen for -8 Bytes!

c=>(n=[],l="",(c+',').split`,`.map(p=>l=a(/[([{]/)>a(/[)\]}]/)?l+','+p:(l&&n.push(l),p)),n);a=r=>l.split(r).length

TIO

Javascript, 140 132 Bytes

Thanks to @KevinCruijssen for -8 Bytes!

c=>{n=[];l="";for(p of (c+',').split(',')){m=l.split(/\(|\[|\{/).length==l.split(/\)|\]|\}/).length;m&&l?n.push(l):0;l=m?p:l+','+p}return n}

TIO

Javascript, 140 132 Bytes

Thanks to @KevinCruijssen for -8 Bytes!

c=>{n=[];l="";for(p of(c+',').split`,`){m=l.split(/[([{]/).length==l.split(/[)\]}]/).length;m&&l?n.push(l):0;l=m?p:l+','+p}return n}

TIO

Javascript, 140 Bytes

c=>{n=[];l="";for(p of (c+',').split(',')){m=l.split(/\(|\[|\{/).length==l.split(/\)|\]|\}/).length;m&&l?n.push(l):0;l=m?p:l+','+p}return n}

TIO

Javascript, 140 132 Bytes

Thanks to @KevinCruijssen for -8 Bytes!

c=>{n=[];l="";for(p of (c+',').split(',')){m=l.split(/\(|\[|\{/).length==l.split(/\)|\]|\}/).length;m&&l?n.push(l):0;l=m?p:l+','+p}return n}

TIO